ECTC Section 2 cover - UAB

UABSchoolofEngineering–MechanicalEngineering–EarlyCareerTechnicalJournal,Volume17 Page33

SECTION 2

Numerical Methods

UABSchoolofEngineering–MechanicalEngineering–EarlyCareerTechnicalJournal,Volume17 Page34

UABSchoolofEngineering-MechanicalEngineering-JournaloftheECTC,Volume17 Page35

Journal of UAB ECTC Volume 17, 2018

Department of Mechanical Engineering The University of Alabama, Birmingham

Birmingham, Alabama USA

SIMULATING THE EFFECTS OF TURBULENCE ON PRESSURE TAP MEASUREMENTS

Jose J Corona Jr. North Carolina Agricultural & Technical State University

Greensboro, NC, USA

Louis C. Chow University of Central Florida

Orlando, FL, USA

Quinn H. Leland Air Force Research Laboratory

Dayton, OH, USA

John P. Kizito North Carolina Agricultural & Technical State University

Greensboro, NC, USA

ABSTRACT The main objective of this paper is to study the effects of

turbulence in collecting pressure tap measurements under high Reynolds number flows. The Reynolds numbers have been determined experimentally for free delivery of a 2-bladed fan to range from its free delivery condition to the shutoff condition respectively, 10540 < 𝑅𝑒( < 33,612. For internal flow, the results indicate that the experimental rig is experiencing turbulence.

The fan performance curve was determined numerically under steady state conditions and follows the trend of the experimental results. The CAD simulations were drawn to represent the actual experiment. The boundary conditions and initial conditions are intended to model inherent properties within the experiment and environmental conditions. These preliminary results obtained from SOLIDWORKS flow simulation provide insight on the physics of the flow within the test rig. Due to the high Reynolds numbers in the pipe, we assume that turbulence exists; furthermore, the flow trajectories and vorticity contours have confirmed such behavior. Also, with the simulation we are able to pinpoint the locations where the vortices are concentrated. These pockets of vortices present a possible problem in accurately measuring pressures at the taps. The key components in the loop, such as the venturi flow meter, butterfly valve and the axial fan, have been our primary focus in this study. The flow characteristics around these items are studied extensively. We were able to find that the butterfly valve introduces wakes when the valve is being closed. Additionally, the venturi flowmeter also can have effects of vorticity at certain fan conditions.

We will expand this work to include higher Reynolds numbers up to about 90,000. The interest in this is to determine if the pressure tap measurements will show greater variations in the pressures.

KEY WORDS: Electro-mechanical Actuators (EMAs), fan performance curves, numerical simulations, turbulence, internal flow

INTRODUCTION This study is a continuation of an already existing

experiment. In that paper, the affinity laws were experimentally confirmed to accurately predict the performance curve of a single fan at various rotational speeds and pressures [1]. Figure 1 demonstrates the current experimental setup used in that paper.

Figure 1. Current experimental rig in the laboratory.

The objective is to enhance the cooling performance of

electro-mechanical actuators used in aerospace applications. Electro-mechanical actuators act as a heat source within a wing-bay; therefore, the test loop has been simulating sub atmospheric conditions and the performance of axial fans under these environmental conditions. According to the literature, fans in a series formation increase the static pressure across n number of fans; on the other hand, for fans in a parallel arrangement the volume flow rate is the parameter that increases [2]. Extensive studies have been conducted for parallel systems around Water


Distributions Systems (WDS). The affinity laws have been studied for various applications to accurately predict the best point of operation [3, 4]. According to Koor et al., they investigated the best efficiency point (BEP) of variable speed pumps (VSPs) using a Levenberg-Marquardt algorithm (LMA); furthermore, they discovered that for identical pumps the efficiency is optimized at equal discharge rates [5]. Borsting et al. modeled various methods of grouping VSP’s and optimization techniques [6]. The automotive industry also is heavily investing time to discover better cooling mechanisms. In Filho et al., they studied how installing parallel fans is beneficial to cooling a system used for buses. The electrical fan system was more efficient, less noisy, more controllable, and easier to maintain [7]. Researchers have worked extensively in understanding the characteristics of axial fans. Chow et al. have performed studies of fan performance curves at various ambient pressures [8, 9]. In evaluating fan performance, computational fluid dynamics (CFD) and experimentation have been combined by many researchers. Lin et al. utilized ANSYS to combine numerical results and experimental results for improving performance of fan design [10]. Additional methods of cooling have been studied. Combining both series and parallel arrangements to cooling heat generation enclosures have been investigated. These studies found that small fans function well for areas of localized heat generation, and large fans dissipate heat effectively in large areas [11]. Other methods of cooling have been proposed, such as using the working principles within heat pipes to transfer localized areas of heat to the ambient for cooling of electrical machines [12].

Using fans to cool the electronic components has the capability of stirring the ambient air within the system. Thus, this mixing can introduce eddies which can enhance the heat transfer within the structure. Studies of turbulent pipe flow have been extensively studied since Osborne Reynolds proposed the idea. Researchers have studied the methods to disrupt turbulence within pipe flow [13]. Other studies have centralized their work on studying the behavior of pipe flow at low Reynolds numbers with the use of various turbulence solvers and experimentation [14, 15].

The high Reynolds number flows within the pipe could potentially provide erroneous pressure measurements at the taps. Xu et al. have performed numerical studies on errors of pressure measurements related to orifice flowmeters [16]. The objective of this paper is to explore whether the pressure measurements have been compromised by the turbulent flow within the system. The physics of the flow within the test loop will provide insight into improving the cooling capabilities.

EXPERIMENTAL RESULTS The fan performance was determined experimentally for the fan operating at standard pressure and temperature conditions while rotating at a constant speed of 20,000 rpm. The performance of a fan can be determined by measuring the pressure differential across the fan and a flow measurement device. We have chosen a venturi flowmeter to determine the volumetric flow rate within the loop. The pressure tap positions are demonstrated in Figure

2; moreover, the taps at the fan are positioned 2’’ before and after the fan. The Nylon was machined to mount the fan into the butyrate tube displayed in Figure 3. The fan performance curve obtained with the AMETEK Rotron two-stage fan is shown in Figure 4. Figure 5 demonstrates the geometry of the AMETEK fan in closer detail.

Figure 2. Pressure taps in the simulation are placed in

identical positions as the experimental taps.

Figure 3. The installation of the two-stage axial fan into

the butyrate tube.

Figure 4. Fan performance curve determined experimentally with a single fan at 20000 rpm.

Figure 5. AMETEK Rotron Propimax2 geometric features.


The equation for the volume flow rate of an obstruction flowmeter can be obtained by combining Bernoulli’s equation with continuity to yield [17, 18],

𝑄 = 𝐶0𝐴234(∆7)

9(:;<=), (1)

where ∆𝑝 is the differential pressure in the throat and upstream of the venturi, 𝜌 is the density, 𝛽 = 𝑑

𝐷C the diameter ratio of throat to upstream diameter, 𝐴2 is the cross-sectional area in the throat of the flowmeter, and 𝐶0 is the discharge coefficient.

A hot-wire anemometer by Testo 405i was used to determine the fluid velocity experimentally. In Figure 6 the results demonstrate rapid fluctuations in the values recorded by this unit.

Figure 6. Demonstrates the rapid fluctuations in fluid

velocity measured with the Testo 405i.

This data was recorded 26 inches downstream of the fan. We encountered interesting results in collecting data at the

shutoff condition of the fan; furthermore, with the use of SOLIDWORKS flow simulation, we will explore those effects. The shutoff condition is experimentally obtained when the flow is constrained with the use of a flow regulating device such as a ball valve or a butterfly valve. For our experiments we selected the latter by NIBCO. Using such a device with rough edges will introduce areas of eddies within the flow.

When the valve is fully closed, the flow is forced to bounce back from the valve. This condition provokes a back-pressure buildup, which could introduce wall shear stresses that could potentially lead to erroneous data collection. In tapping holes for pressure measurement, it is important to properly deburr the area since it may lead to flow disturbances.

CAD DESIGN OF THE LOOP In this paper, the experimental loop shown in Figure 1 was

modeled in SOLIDWORKS to closely represent the effects within the testing loop. The turbulence model that is used in SOLIDWORKS flow simulation is the Lam-Bremhorst k-e turbulence model.

Figure 7. CAD design of the loop.

The source term was placed at the red region of the

simulation, and the blue box indicates the location of the butterfly valve shown in figure 7. The two-stage axial fan is located 11.75’’ downstream of point one, shown in the Figure 7. In the simulation, the rotational effects of the fan were modeled, and those results are observed in the flow trajectories shown in the following sections of this manuscript. The valve was simulated for four different positions in steady state, and up to this point has been simulated for two positions under transient conditions. Additionally, there is a flow straightener downstream of the fans. This straightener is located 2.5 inches upstream of the venturi flowmeter.

Figure 8. Flow straightener used in the simulation.

Figure 9. CAD of the venturi flowmeter used for

simulation results. The drawings provided in Figures 8 and 9 have been carefully constructed to meet specific sizes of the physical components in the test loop. The venturi flowmeter has been constructed in-house and follows the specifications of machine venturi from ISO-5167 [19].


BOUNDARY AND INITIAL CONDITIONS The preliminary numerical results obtained in this study

were conducted through SOLIDWORKS Flow Simulation software. These preliminary results served to provide initial thoughts on the physics of the flow within the loop.

Steady state, gravitational effects in the y-direction, standard temperature and pressure within the loop, adiabatic conditions from the wall of the PVC to the environment, roughness of the inner wall for PVC, and both laminar and turbulent flow regimes were the conditions considered for the simulation. Figure 10 demonstrates the boundary conditions assigned to simulate the environment within the test loop.

Figure 10. Demonstrates the boundary conditions used to model the behavior within the loop.

The red squares show the roughness assigned the wall (𝜀 =0.0015𝑚𝑚), the blue arrows at the bottom are the pressure ambient pressure conditions (𝑃 = 101325𝑃𝑎), and the green arrows are the momentum source term of the fan.

The simulation was conducted for four different valve positions. Those valve positions were simulated by rotating the bottom center circle about the x-z plane at 90°, 60°, 30°, and 0°. The 90° is the fully open condition (i.e. free delivery) and the 0° being the fully closed condition or shutoff.

The source term was specified by providing SOLIDWORKS with the fan performance curve indicated in Figure 2. In the simulation we are considering the fan to already have been initiated to 20,000 rpm at time 𝑡 = 0𝑠.

GRID INDEPENDENT STUDY The model was tested for grid independence. To guarantee

that the model is grid independent, a single valve position was tested with various mesh sizes provided in SOLIDWORKS. The mesh size was studied extensively, varying with differential meshing within the model and uniform meshing within the model. Initial results indicated that a denser mesh was required in the throat of the venturi to ensure consistent data. The denser the mesh, the more confident one can be in the results provided in the simulation; therefore, a differential grid sizing of 10;L𝑚 and 10;M𝑚 was taken as the most accurate result in the loop and throat of the venturi, respectively. The percent difference was calculated amongst the various mesh sizes and given in Table 1.

Table 1. The percent difference between the different grid sizes compared to the densest mesh.

Pressure [Pa]

Non-uniform Grid

Uniform Grid Size 1mm

Uniform mesh 0.5mm

PG1_3 0.003903926 0.001454446 2.9104E-08 PG2_3 0.054040662 0.016276622 1.12067E-08 PG3_3 0.02231483 0.022195182 1.06329E-08 PG4_3 0.138913646 0.040686807 4.74084E-08

According the results of Table 1, the percent difference of

the pressures between the different mesh settings is less than 1%; consequently, that indicates that the study was grid independent. The remainder of the valve settings were simulated at a mesh size with uniform grid size of 10;L𝑚 to save computation time.

TIME STEP INDEPENDENCE To test if the numerical results are independent of the time

step, we considered different time steps of ∆𝑡 = 0.1𝑠 and ∆𝑡 =0.01𝑠 for a total time of 𝑡 = 0.5𝑠. The test loop condition considered was at free delivery of the fan and standard pressure and temperature conditions within the loop. Table 2. Demonstrates the time step independence of the

simulation. Parameter ∆t=0.1s ∆t=0.01s %Difference

PG1_3 101303.7 101304.4 7.16739E-06 PG2_3 101498.6 101509.8 0.000109894 PG3_3 101414.3 101429.6 0.000151266 PG4_3 101157.8 101189.6 0.000314195

The size of the time steps demonstrates similar results for

the two cases studied; therefore, the results obtained in the transient results section will be conducted at the smallest time step in table 2. Preliminary results were obtained at ∆𝑡 =0.001𝑠, but will be further tested to verify that we are capturing the physics of the flow and that the results presented here are accurate.

STEADY STATE RESULTS First, we considered the flow within the system to be

independent of time. The flow trajectories within the loop can provide insight as to what the flow characteristics appear to be in the experimental rig. The simulation was tested in the case where the valve is fully open to allow minimal flow resistance within the loop.


Figure 11. Flow trajectories within the loop simulating

free delivery of the fan. Figure 11 above shows that the greatest velocity is seen at the throat of the venturi, and that the flow becomes twisted as soon as it passes through the fan. The flow straightener has the ability of eliminating the rotational characteristics of the flow; but due its small length in the axial direction, the flow becomes twisted shortly after, as demonstrated in Figure 12.

Figure 12. Demonstrates the flow straightener’s effect on

the fluid particles. Perhaps including a second flow straightener in the loop at

a set distance from the first could disrupt the flow and make it straighter.

Next, the closed valve system was simulated to study the flow particles in the loop. The steady state result yielded that enough time has been allotted to the system. This indicates that the pressure has built up significantly, and the flow has become stagnant within the system, which is shown in figure 13.

Figure 13. Flow trajectories of the loop at the shutoff

condition. The highest velocity determined in the simulation is higher

in this case than the case mentioned previously. Furthermore, the highest velocity is located in the region around the fan. This

behavior can be explained by the fact that the flow has stagnated and is continuously being supplied with kinetic energy by the fan.

Recall that the pressure taps of the loop are labeled in Figure 2. The difference of PG1 and PG2 will provide the static pressure rise of the fan, and the differential pressure of the PG3 and PG4 will give the volumetric flow rate. Due to time restrictions, the fan performance curve was not obtained at a uniform mesh of 1mm, but initial results were obtained with the coarser, non-uniform mesh.

Figure 14. Fan performance curve that compares the

experimental and SOLIDWORKS flow simulation results.

The results demonstrate that the numerical results closely approximate the experimental results at low volumetric flow rates. On the other hand, the higher volume flow rates show greater differences in the data among the two scenarios shown in Figure 14. Lam-Bremhorst turbulence models are typically utilized for low-Re flow simulations; nonetheless, if the grid is carefully accounted for then the simulations can be conducted with this turbulence model even at high Reynolds numbers.

TRANSIENT STATE RESULTS We consider the transient effects in the loop mainly for the

condition of the fully closed butterfly valve; furthermore, under this condition, the experimental results demonstrate complications. These difficulties in collecting the data could be as a result of flow reversals in the loop causing undesirable shear stress at the wall taps. The transient state results were tested taking the time step to be 0.01s, the results were tested for a duration of 1s. The following graphs demonstrate the pressure at the wall for the different locations of the point measurements in the testing loop. The results yield that a minimum of 0.45s should be given for startup time. Once that time has elapsed, the pressure readings are steady across every point where the pressure is determined experimentally.


Figure 15. Comparison of the numerical and experimental

pressures at the start.

From Figure 15, the comparison between the numerical and experimental pressures are demonstrated from the start of the test runs. The data acquisition system in the lab has the ability of obtaining five points per second, which are indicated by the dashed lines; on the other hand, SOLIDWORKS flow simulation has the capacity a much higher rate of values within that same time frame. All the numerical results overestimate the experimental values by 155 Pa, 189 Pa, 149 Pa, and 242 Pa for points PG 1, PG 2, PG 3, and PG 4, respectively. The figure also demonstrates that numerically, the pressure achieve a convergence point past approximately 45 seconds. The experimental and numerical data shown in figure 15 are at the same conditions (ie. 20000 rpm and free delivery condition).

The Reynolds number within the system varies as the butterfly valve is rotated to restrict the flow. The higher Reynolds number shows a higher volumetric flow rate, and that is shown in Figure 16 to be consistent throughout the time elapsed in the plot. Additionally, the difference is widened as time progresses. Nonetheless, both conditions still demonstrate that more time needs to progress for steady state results.

Figure 16 demonstrates that the higher Reynolds number would require more time for steady state results.

The numerical results obtained from SOLIDWORKS flow simulation demonstrate that after approximately 0.45s, the solutions demonstrate steady state behavior for the fully open valve condition. Therefore, we are confident in our experimental results for this particular fan curve since the Reynolds numbers are in a fully turbulent regime. Further analysis for higher Reynolds numbers will be tested to determine the time necessary for steady state results.

The transient effects are to be closely examined for when the valve is fully closed. The experimental results show that

complications have been encountered when the flow is obstructed by the valve. The smallest time step did not demonstrate ground breaking results when the fan is operating at the shutoff condition. The trajectories demonstrated similar results of stagnant flow conditions due to the back-pressure buildup.

Figure 16. Volumetric flow rates determined by

SOLIDWORKS at different Re when butterfly valve is fully closed.

Simulations were conducted when the system was in 30° and 60° flow resistances. Intermediate valve positions have presented complications in the experimental results, especially when very minimal flow is allowed through the system. The current results have been tested for a time of 𝑡 = 0.1𝑠. This time was chosen due to long computation times to obtain a solution. The geometry of the loop is very large, and due to the grid size combined with transient effects the run times are significant. The flow trajectories demonstrate a chaotic behavior. The velocity of the particles increases around the edges of the valve and the wall. Along the right side of the valve, particles tend to follow a straight path. On the opposite side of the valve the flow tends to recirculate, shown in Figure 17. Figure 18 demonstrates a snapshot of the vorticity around the valve. The vorticity has rapid fluctuations in the small time that this parameter was studied. Allowing for a greater run time would reveal whether the fluctuations are to eventually vanish with time.

The high vorticities around the valve indicate that the flow has patches of strong rotation that is introduced by the valve. The vortices are greater near the wall and directly behind the valve.

A similar analysis was performed on the venturi to understand the flow in that section of the loop. Along the top the high vorticity is due to the fan’s kinetic energy stirring up the flow. As the flow goes downstream and time progresses, the vortices are translated towards the center of the pipe. The frictional effects at the wall cause the vortices to dissipate near the wall downstream of the fan. When the flow interacts with the straightener, vortices begin to form near the edges and the solid structure of the straightener. Nonetheless, those are effects are negligible compared to the results shown in Figure 10. The change in diameters from upstream of the venturi and in the throat causes vortices to form.


Figure 17. Flow trajectories around the valve when

halfway open.

Figure 18. Vorticity around the valve when simulated at

the midway position at 𝑡 = 0.07𝑠.

Figure 19. Isometric view of the vortices in the top section

of the loop.

Taking a smaller time step would demonstrate more gradual results in the vortices along Figure 19. Regardless, having such a small-time step will not aid our results in properly measuring the pressure at the taps since the results tend achieve steady state behavior after approximately half a second.

CONCLUSIONS The simulations showed that the results achieve steady state

behavior approximately at 𝑡 > 0.45𝑠 for the fully open butterfly valve condition. This gives a strong indication that when taking the pressure measurements at the taps, we can be assured that the results are accurate at any point of the experiment.

The fan performance curve was determined numerically under steady state conditions, and follows the trend of the experimental results. The CAD simulations were drawn to mimic the actual experiment in every aspect. The boundary conditions and initial conditions are intended to model inherent properties within the experiment and environmental conditions within the testing facility. These preliminary results obtained from SOLIDWORKS flow simulation provide insight on the physics of the flow within the test rig. Due to the high Reynolds numbers in the pipe, we assume that turbulence exists; furthermore, the flow trajectories and vorticity contours we have confirmed our hypothesis. Also, with the simulation we are able to pinpoint the locations where the vortices are concentrated. These pockets of vortices present a possible problem in accurately measuring pressures at the taps. We were able to explore the dynamics of the fluid particles around key components in the loop. We were able to find that the butterfly valve introduces wakes when the valve is being closed.

We will expand this work to include higher Reynolds numbers up to about 90,000. The interest in this is to determine if the pressure tap measurements will show greater variations in the pressures at higher Reynolds numbers, and if the venturi flow gives accurate results for volumetric flow rate.

REFERENCES [1] Corona, J. and Kwarteng, A. “Testing of a Cooling Fan for Wing-bay Electro-Mechanical Actuators.” 5th Joint US-European Fluids Engineering Division Summer Meeting. ASME, no. FEDSM2018-83480 (2018). [2] Wright, Terry. 1999. Fluid machinery: performance, analysis, and design. Boca Raton, Florida: CRC Press. [3] Augustyn, Ockert P. H., Sybrand J. van der Spuy, and Theodor W. von Backström. "Numerical and Experimental Investigation into the Accuracy of the Fan Scaling Laws Applied to Large Diameter Axial Flow Fans." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 230, no. 5 (2016): 477-86. [4] Marchi, A. and Simpson, A.R. "Evaluating the Approximation of the Affinity Laws and Improving the Estimate of the Efficiency for Variable Speed Pumping." Journal of Water Resources Planning and Management 139, no. 12 (2013): 1314-17. [5] Koor, M., A. Vassiljev, and T. Koppel. "Optimization of Pump Efficiencies with Different Pumps Characteristics


Working in Parallel Mode." Advances in Engineering Software 101 (2016): 69-76. [6] Borsting, Zhenyu Yang and Hakon. "Energy Efficient Control of a Boosting System with Multple Variable-Speed Pumps in Parrallel." 49th IEEE Conference on Decision and Control (December 15-17, 2010 2010). [7] Ivar José de Souza Filho, Adriano de Oliveira Francisco, Bruno Scarano Paterlini, Daniel Luna Ferreira da Silva, Marcello Cervini Procida Veissid, Paschoal Federico Neto, and Ricardo Moreira Vaz. "Application Study of Electrical Fans Assemble Applied in Bus Cooling System." SAE International 24th SAE Brasil International Congress and Display, no. 0148-7191 (2015). [8] Wu, Wei, Yeong-Ren Lin, Louis C. Chow, and Quinn Leland. "Fan Performance Characteristics at Various Rotational Speeds and Ambient Pressures." In SAE Technical Paper Series, 2014. [9] Wu, Wei, Yeong-Ren Lin, Louis Chow, Edmund Gyasi, John P. Kizito, and Quinn Leland. "Electromechanical Actuator Cooling Fan Blades Design and Optimization." In SAE Technical Paper Series, 2016. [10] Lin, Sheam-Chyun, Shen Ming-Chiou, Tso Hao-Ru, Yen Hung-Cheng, and Chen Yu-Cheng. "Numerical and Experimental Study on Enhancing Performance of the Stand Fan." [In English]. Applied Sciences 7, no. 3 (2017). [11] al., Barron et. "Parrallel-Series Hybrid Fan Cooling Apparatus and Optimization." no. US 2017/0082112 A1 (2017). [12] Kabir, Rifat, Patrick McCluskey, and John P. Kizito. "Investigation of a Cooling System for a Hybrid Airplane." In 2018 AIAA/IEEE Electric Aircraft Technologies Symposium, p. 4991. 2018. [13] Kühnen, Jakob, Baofang Song, Davide Scarselli, Nazmi Burak Budanur, Michael Riedl, Ashley P. Willis, Marc Avila, and Björn Hof. "Destabilizing Turbulence in Pipe Flow." Nature Physics 14, no. 4 (2018): 386-90. [14] Avila, Marc, Ashley P. Willis, and BjÖRn Hof. "On the Transient Nature of Localized Pipe Flow Turbulence." Journal of Fluid Mechanics 646 (2010). [15] Kerstin Avila, David Moxey, Alberto de Lozar, Marc Avila, Dwight Barkley, Bjorn Hof. "The Onset of Turbulence in Pipe Flow." SCIENCE 333 (8 July, 2011 2011): 192-96. [16] YinQin Xu, Daniel Coxe, Yulia Peet, Taewoo Lee, . "Computational Modeling of Flow Rate Measurements Using an Orfice Flow Meter." American Society of Mechanical Engineers FEDSM2018 (2018). [17] White, Frank M. 1999. Fluid mechanics. Boston, Mass: WCB/McGraw-Hill. [18] Cengel, Yunus A., and John M. Cimbala. 2006. Fluid mechanics: fundamentals and applications. Boston: McGraw-Hill Higher Education. [19] ISO, "Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full— Part 4: Venturi tubes", (2003), Vol. ISO 5167-4:2003(E);





NUMERICAL PREDICTIONS OF HYDROKINETIC TURBINE WAKE AND FREE-SURFACE SIGNATURES

O. El Fajri, S. Bhushan Mississippi State University Starkville, Mississippi, USA

D. Thompson Mississippi State University Starkville, Mississippi, USA

ABSTRACT The flow associated with a hydrokinetic turbine (HKT) is

more complex than that for wind turbines. Its complexity lies in the interaction with the air-water interface, which can both affect wake recovery and result in signatures on the surface. Computational Fluid Dynamics (CFD) has proven to be a well-established method for wake evaluation in terms of cost and time compared to experiments. Previous numerical studies have focused on the wake recovery of an isolated HKT in single phase flow using ANSYS/Fluent. Herein, we will extend the study and focus on the free surface effect on the HKT wake and reciprocally the effect of the wake on the free surface. The results and details of the wake recovery will then be validated using experimental data from literature and previous single-phase Fluent simulations. The study demonstrates that OpenFOAM results were in good agreement with Fluent results. The thick boundary layer accelerates the flow and creates a blockage effect. Hexahedral meshes perform better than mixed cells grids by displaying well-defined vortices structures, and DES shows more resolved turbulence. For two-phase flows, slight signatures are visible at the surface. HKT rotation induces wave elevation at the air-water interface.

KEY WORDS: Hydrokinetic turbine, wake, recovery, free surface.

INTRODUCTION The majority of greenhouse gas (GHG) emission is due to

the consumption of fossil fuels. Renewable energy sources such as bioenergy, solar energy, geothermal energy, wind energy, and hydropower are a viable alternative to lower GHG emissions. Unlike wind energy which is a conventional and popular energy harvesting method, hydropower offers a high potential for energy extraction, since water represents 71% of the earth’s surface. Different approaches and techniques are developed for different resources such as wave, tidal, and ocean currents [1]. Hydrokinetic energy extracted from flowing currents using hydrokinetic turbines (HKT) produces electricity, which can be

used to power cities. The wind turbine is a well understood technology [4, 5] and it shares some issues with HKT.

HKT studies focus mainly on two areas: the rotor performance and the wake recovery. This work focuses on the wake region behind an individual turbine. The results and conclusions extracted from a single turbine are essential to HKT farm design. Model scale experimental studies were carried out by Eggar et al. [2] and Maganga et al. [3] to understand the HKT wake characteristics. However, experimental investigation to analyze the array configuration and turbine geometry is extremely expensive. Therefore, computational fluid dynamics (CFD) serves as a reasonable alternative for a thorough evaluation of the hydrokinetic turbine. The HKT wake is divided into three regions: the near wake region located up to 1D downstream (diameter) which is related to the turbine blades’ characteristics and power extraction [6], the intermediate wake region located between 1D and 5D, where we can evaluate the vortices generated by the turbine blades, [7] and the far wake region beyond 5D ,which focuses mostly on turbulence modeling and wake evolution [8].

Numerical strategies to model hydrokinetic turbine wake with accuracy range from the inexpensive actuator disk approach [18] to the computationally expensive large eddy simulation (LES) blade resolved turbine [19]. The blade resolved model is computationally expensive and is categorized into three types: the single reference frame, where the domain rotates with the turbine [4], the moving reference frame (MRF), where the turbine doesn’t rotate but the grid volume surrounding the turbine rotate; and finally, the rotating frame model with two domains, the rotating and stationary region communicating using a sliding interface.

The aim behind using different turbulence models for turbine wake prediction is to evaluate turbulence accurately. Kasmi et al. [9] and Cabezon et al. [10] used the Reynolds Averaged Navier Stokes (RANS) model and noted the underestimation of the near wake turbulence. Salunkhe et al. [11] performed Fluent simulations for a 0.5 m diameter HKT using a resolved rotating blade model with unsteady RANS (URANS), LES, and detached eddy simulation (DES) turbulence models. The results were validated against the available experimental


data. The primary conclusions and limitations identified by the study [11] were:

(1) The pressure distribution on the blades is the principal source of power and thrust. The latter are independent of turbulence models and the predictions agree with 4.3% of the experimental data.

(2) Fine grid resolution (~ 8.8M cells) is necessary for accurate intermediate wake prediction. The turbulence model choice has an impact on the wake prediction. LES performs better than URANS and IDDES for turbulence anisotropy with 7% of averaged prediction errors. It also underpredicted the wake diffusion and exhibited a difference in the wake deficit pattern which could be due to the negligence of free surface effects.

(3) The numerical results identified some limitations concerning the intermediate wake turbulence prediction and deficit, which could be due to an interface boundary condition issue and mesh generation approach.

The objective of the present work is to perform turbulent simulations using OpenFOAM for an isolated model-scale, three-blade HKT, with resolved rotating turbine model and URANS and DES turbulence models with different grid types namely: mixed and hexahedral cells grids generated by Pointwise and SnappyHexMesh (SHM). To further the investigation on the CFD limitations identified in Salunkhe et al. [11], OpenFOAM results will be compared with Ansys/Fluent results and validated against experimental data. Additionally, we will investigate the effect of the side walls boundary layer on the flow using the different mesh generation methods. Next, the study will be extended to two phase flows where we will focus on the free surface effect on the wake growth and reciprocally; to get a better understanding of the interactions at the air-water interface.

EXPERIMENTAL DATA Model-scale and full-scale experimental tests were

performed for different turbine types to assess the wake recovery mechanism and the power output. Table 1 summarizes some of the available experimental data for a horizontal axis tidal turbine (HATT). Egarr et al. [2] examined a 4 blade with Wortmann FX 63-137 profile HATT of 5.5 m diameter to determine the optimum power extraction from the tide. Morris [13] investigated the performance, swirl and wake characteristics for 0.5 m diameter turbine. Maganga et al. [3] conducted experiments for a 3-blade turbine of 0.7 m diameter with different yaw angles. The study concluded that wake recovered faster at higher turbulence intensity (TI) levels. Tedds et al. [14] carried out experiments on a 3-bladed turbine of 0.5 m diameter with stanchion. The study concluded the negligence of swirl effects in previous approaches over-predicted the turbulence kinetic energy (TKE) decay rate. Other flume and wave tank experiments for a 3-bladed model scale HKT are ongoing at IFREMER.

NUMERICAL METHODS The simulations were performed using OpenFOAM, the

open-source computational finite volume CFD toolbox developed by Jasak et al. [15]. PimpleDyMFoam was the transient solver used for incompressible, turbulent flow of Newtonian fluids on a moving mesh. Robertson et al. [12] have validated OpenFOAM numerical methods and turbulence models for incompressible bluff body flows in single phase, which includes flow over a backward facing step, sphere and sharp leading-edge delta wing. The study identified the 2nd order linear upwind and 1st, 2nd order blended limited linear schemes as the most efficient for RANS and hybrid RANS/LES (HRL) simulations, respectively.

The flow fields are governed by the incompressible Navier-Stokes equations. Turbulence models assume a decomposition of the instantaneous velocity (ui) into resolved (𝑢"#) and modelled (𝑢′#) components:

𝑢# = 𝑢"# + 𝑢′# (1)

where velocities are defined in the Earth-fixed reference frame, and i represents the x, y and z directions, respectively. Application of the filtering operation to the Navier-Stokes equations yields,

'()*'+*

= 0 (2a)

'()*'-+ .𝑢"/

0 '()*'+12 = − 4

567"6+*+ 𝜈 '9()𝒊

'+1'+1−

';*1'+1

(2b)

The 𝜏#/ term represents the turbulent stresses:

𝜏#/ = 𝑢=𝑢>? −𝑢"#𝑢"/ (3)

The turbulent stresses are modeled based on the Boussinesq assumption for isotropic turbulence as,

𝜏#/ = 𝜈A𝑆#/ (4)

where, 𝜈A is the turbulent eddy viscosity and 𝑆#/ is the rate-of-strain tensor.

The URANS blade-resolved simulations were performed using the k-omega shear stress transport (SST) [16] and DES [20]. Six model-scale simulations of a 3-blade, resolved, rotating model turbine of 0.5 m diameter with and without stanchion were performed and are listed in Table 2. For an inflow velocity of Uo=0.892 m/s at λ=6.15 (ω=21.932 rad/s), the turbine is submerged in water with a density of ρ=998.2 kg/m3 and with a Reynolds number of ReD=2.2x105. The TI was set to 2% for all simulations.

Figure 1 represents the computational domain used for these simulations. The turbine is located at (0,0,0) streamwise, spanwise and transverse respectively. The domain extends 5D upstream and 21D downstream of the turbine, y = [-0.75D, 1.15D] spanwise and z = [-1.35D, 1.35D] transverse.


Table 2. Details of the simulations performed in the study

Runs Stanchion Grid Turbulence

model Flow phase Size Type Wall

y+ HKT y+

Fluent FL(U) Yes 8.8 M

Mixed hex and Tets

500 42 URANS Single FL(L) LES

Open FOAM

OF-MC1(U) Yes 8.9M 290 3.8 URANS Single

OF-MC2(U) Yes 11 M 1 4.5

URANS Single OF-

MC2(D) DES

OF-SHM(U) Yes 7M

Hex 394 1

URANS Single OF-

SHM(D) DES

OF-SHM-2P(U)

No 12M URANS Two

Figure 1. Computational domain consisting of two blocks – stationary and rotating block showing the boundary

conditions and directions. For the cases OF-MC, the mixed-cell mesh (hexahedral-

tetrahedral-prism) was generated using Pointwise. T-REX (anisotropic tetrahedral extrusion) grid was used with hexahedral cells in the turbine boundary layer and tetrahedral cells for the rest of the domain as shown in Figure 2. For OF-MC1 the y+ value was not fixed near the domain walls; oppositely for OF-MC2 the y+ value near the walls was fixed to 1. The y+ variance will help evaluate the effect of the boundary layer developing on the walls on the flow.

Figure 2. Cross-sectional view of the hybrid 11M cells grid at x/D = 0 plane showing the wake refinement region and the nature of the cells used generated using Pointwise.

OF-SHM and OF-SHM-2P meshes were generated using

OpenFOAM mesh utility SnappyHexMesh, which does not have a body fitted grid but has cut-cells. OpenFOAM requires a hexahedral mesh for stability. It is not ideal for resolving the boundary layer, but it is reasonable for primarily pressure driven flows such as HKT. The Figure 4 flow chart summarizes the steps to follow to generate an adequate mesh for a resolved rotating blade model using SnappyHexMesh. It offers more flexibility for mesh specification such as refinement and mesh control parameters. The output mesh corresponds to the hexahedral grids as shown in Figures 3-5. The objective behind using two different mesh generation approaches is to evaluate the grid resolution’s impact on wake characteristics.

For the preliminary free surface study, the geometry was simplified by removing the stanchion. This case requires more refinement at the free surface location to capture all interactions happening at the air-water interface; therefore, a refinement region was added as shown in Figure 5.


Figure 3. Cross-sectional view of the hex 7M cell grid at

x/D = 0 plane generated using SnappyHexMesh.

Figure 4. Flow chart showing the requirements to generate a mesh using SnappyHexMesh utility.

Figure 5. Cross-sectional view of the hex 12M cells grid at

x/D = 0 plane for the free surface case generated using SnappyHexMesh.

Figure 6b-c-d represents the contour plots of the mean streamwise velocity for cases OF-MC1(U)-MC2(U)-SHM(U). In Figure 6b, the boundary layer (BL) developing on the side walls is thick and starts early upstream of the domain, which can create a blockage effect for the incoming flow. The BL is helping the flow accelerate at z/D=± 0.6-0.8. After fixing the y+ value to 1 in Figure 6c, we can note that the flow acceleration has reduced at the blade tip locations. For Figure 6a-d, the boundary layer on the walls is not apparent. These contour plots suggest that the wall function was not effectively implemented for the mixed cells grid in OF-MC1-2.

a) FL(U) b) OF-MC1(U)

c) OF-MC2(U) d) OF-SHM(U)


e) Streamwise velocities at x/D=1.5

Figure 6. Streamwise velocity contour plot of three

OpenFOAM simulations using URANS at y/D=0 showing the boundary layer on the side walls.

From Figure 7a-b-c-d, we can identify different types of

vortex structures: The tip vortex created at the tip of the rotating blades, and the root vortex generated behind the hub of the HKT. The vortex rings are destroyed as they reach the stanchion. The root vortex decays faster than the tip vortex. Figure 7c-d shows sharp tip vortex rings, which are due to the grid cell difference. We can also note the contrast in the vortex structures and extent for the different turbulence models. For the case of a mixed cells grid as shown in Figure 7a-b, the structure does not differ from URANS to DES. However, in Figure 7d DES demonstrates more turbulence structures than URANS. This figure shows how grids generated using the SnappyHexMesh utility provide well-defined structures.

a) OF-MC2(U) b) OF-MC2(D)

c) OF-SHM-(U)

d) OF-SHM-(D)

Figure 7. Vorticity magnitude contour plots of three OpenFOAM simulations using URANS and DES. The streamwise and transverse mean velocity contour plots

for the experimental and CFD simulations are shown in Figures 8-9. In Figure 8, the peak wake velocity deficit for all the contours is located in the near wake region right behind the turbine blade tips corresponding to x/D=1.5 around z/D=±0.5. The wake continues to develop further downstream until it reaches the center by x/D=4.5 for the experiment, and x/D=5.5 for OF-MC2(U). The peak deficit is due to momentum extraction by the rotating blades. For Figure 8d, the peak velocity deficit located at z/D=±0.5 is curved inwards toward the center of the wake and extends downstream. The numerical simulation shows consistent predictions. Figure 8e-f-g shows higher streamwise velocity deficit behind the blades. The transverse mean velocity contour plots in Figure 9 display positive and negative values for the velocity, which are due to the swirl generated by the rotating turbine blades. Figure 9d, exhibits the velocities extending further downstream and reduced in width compared to Figure 9c. The negative transverse velocity in Figure 9a converges to the center line at the opposite of Figure 9b-9c-9d where the transverse velocities maintain the symmetry. Figure 9e shows a decrease of peak value. But in Figure 9g the profile is asymmetric, shifting towards z/D=0 for the positive velocity.

a) Experiment


b) FL(U)

c) OF-MC2(U)

d) OF-SHM(U)

e) FL(L)

f) OF-MC2(D)

g) OF-SHM(D)

Figure 8. Streamwise mean velocity contour plots of the wake at plane y/D=0 using URANS and DES are

compared with experimental data.

a) Experiment

b) FL(U)

c) OF-MC2(U)

d) OF-SHM(U)

e) FL(L)

f) OF-MC2(D)

g) OF-SHM(D)

Figure 9. Transverse mean velocity contour plots of the wake at plane y/D=0 obtained using URANS and DES

are compared with experimental data. The mean velocity profiles predicted by the CFD

simulations are compared with the experiment in Figure 10. Figure 10a, shows the mean streamwise velocity profiles. OF-SHM(D) demonstrates a high deficit at x/D=1.5. FL(U-D), OF-MC2(U-D) and OF-SHM(D) profile results maintain a gaussian wake profile as we go further downstream. However, the OF-SHM(U) profile shape does not change throughout the wake region. Figure 10b, shows the peak transverse velocity is located around z/D=±0.5, and it decreases toward the centerline. The


OF-SHM(D) profile shows the highest deficit in the near and intermediate wake region. The transverse velocity magnitude decreases as we go further downstream.

a) Streamwise velocity line plots

b) Transverse velocity line plots

Figure 10. Streamwise and transverse velocity line plots obtained using URANS and DES are compared with

experimental data. Figure 11 represents the sum of pressure and viscous forces

predicted on the turbine versus the number of blade rotation cycles. It demonstrates unsteadiness for all turbulence models. OF-MC2 shows higher force magnitudes (~90N). OF-SHM predicts lower forces, which could be due to the grid resolution.

Figure 11. Forces VS Time history predictions obtained

using URANS and DES cases. The following figures represent the preliminary test case for

the two-phase flows study for HKT without stanchion. Figure 12 shows the wake generated behind the turbine blades as well as an acceleration of the flow at the air-water interface location which extends to 4D downstream. Figure 13 demonstrates the interactions happening at the air-water interface along with the vortices generated by the turbine. The turbine rotation affects the air-water interface by creating dips downstream and waves upstream of the domain. The turbulence structures generated by the turbine elevate the water level. These preliminary signatures at the air-water interface show that the free surface can influence the flow.

Figure 12. Streamwise velocity contour plot with free

surface location of HKT without stanchion using URANS.

Figure 13. Wave elevation contour plot and turbulence vortices generated by the turbine.

CONCLUSION The objective of this study was to perform initial

OpenFOAM numerical simulations for a 3 blade HKT using a resolved rotating blade model with different grid generation methods, to compare between Fluent and OpenFOAM results, to investigate the blockage effect of the side walls boundary layer, to evaluate the grid topology effect, and to extend the study to two phase flow. OpenFOAM results show a qualitative agreement with the Fluent results in terms of wake recovery.

The specification of the y+ on the walls has an impact on the development of the side wall boundary layer and therefore the flow dynamics. For mixed cell grids, the thick boundary layer creates a blockage effect and accelerates the flow. The wall function does not operate well with high y+ value on the walls for the mixed cells grid. On the other hand, a low value of y+ reduces the thickness of the BL and gives more consistent results with Fluent. For the hexahedral grid, the SnappyHexMesh utility displayed well-defined vortex structures. DES results demonstrate more well-defined turbulence vortices extending farther downstream. Herein, the wall function is effective even with high values of y+ on the walls. The accuracy of SHM can be addressed by using y+=1 on the walls, which will be investigated later.

The preliminary two-phase flows demonstrated dips and wave elevation at the air-water interface caused by the turbine blades’ rotation. The return of the flow toward the inlet can be avoided by extending the domain further upstream, and the

73

75

77

79

81

83

85

87

89

91

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Forc

es (N

)

Blade number of rotation cycles

OF-SHM(U)OF-MC2(U)OF-MC2(D)OF-SHM(D)


signatures at the free surface can be made more prominent by adding the stanchion, which will be investigated later.

ACKNOWLEDGEMENTS This effort was partially supported by the Center for

Advanced Vehicular Systems. All simulations were performed on Shadow HPC system at High Performance computing Collaboratory, Mississippi State University.

REFERENCES [1] Kumar, A., et al. Hydropower. In IPCC Special Report on Renewable Energy Sources and Climate Change Mitigation [O. Edenhofer, R. Pichs-Madruga, Y. Sokona, K. Seyboth, P. Matschoss, S. Kadner, T. Zwickel, P. Eickemeier, G. Hansen, S. Schlömer, C. von Stechow (eds)], Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2011. [2] DA. Egarr, T. O’Doherty, S. Morris, and RG. Ayre, “Feasibility study using computational fluid dynamics for the use of a turbine for extracting energy from the tide,” In 15th Australasian Fluid Mechanics Conference, The University of Sydney, Sydney, Australia, 2004. [3] Maganga, F., Germain, G., King, J., Pinon, G., & Rivoalen, E. Experimental characterisation of flow effects on marine current turbine behaviour and on its wake properties. Renewable Power Generation, IET, 4(6): 498-509, 2010. [4] Sørensen JN. and Shen, W. Numerical modeling of wind turbine wakes. J Fluids Eng; 124: 393-402, 2002 [5] Vermeer, LJ., Sørensen, JN. and Crespo, A. Wind turbine wake aerodynamics. Progress in Aerospace Sciences, 39(6): 467-510, 2003. [6] Shen, W., Mikkelsen, R., Sørensen, JN. and Bak, C. Tip loss corrections for wind turbine computations. Wind Energy 8(4): 457-475, 2005. [7] Elvira, RG., Crespo, A., Migoya, E., Manuel, F. and Hernandez, J. Anisotropy of Turbulence in Wind Turbine Wakes. J. Wind Eng. Ind. Aerod , 93: 797–814, 2005. [8] Crespo A, Frandsen S, G!omez-Elvira R, Larsen SE. Modelization of a large wind farm, considering the modification of the atmospheric boundary layer. In: Petersen EL, Jensen PH, Rave K, Helm P, Ehmann H (editors). Proceedings of the 1999 European Union Wind Energy Conference, Nice, France, 1999.

p. 1109–13. [9] Kasmi, A. and Masson, C. An Extended k-ε Model For Turbulent Flow Through Horizontal Axis Wind Turbines. Journal of Wind Engineering & Industrial Aerodynamics, 96: 103-122, 2008. [10] Cabezon, D., Migoya, E. and Crespo, A. Comparison of turbulence models for the Computational fluid dynamics simulation of wind turbine wakes in the atmospheric boundary layer,” Wind Energy, 14: 909-921, 2011. [11] Salunkhe, S., Bhushan, S., Thompson, D., O’Doherty, D. and O’Doherty; T. Validation of Hydrokinetic Turbulent Wake Predictions and Analysis of Wake Recovery Mechanism.Submitted in Renewable Energy, 2017. [12] Roberston, E., Choudhury, V., Bhushan, S. and Walters, D.K. Validation of OpenFOAM Numerical Methods and Turbulence Models for Incompressible Bluff Body Flows. 2015. [13] Morris, C. Influence of Solidity on the Performance, Swirl Characteristics, Wake Recovery and Blade Deflection of a Horizontal Axis Tidal Turbine, PhD thesis. Cardiff University, UK. 2014. [14] Tedds, SC., Owen, I. and Poole, RJ. Near-wake characteristics of a model horizontal axis tidal stream turbine. Renewable Energy, 63: 222-235, 2014. [15] Jasak, H., Jemcov, A., and Tukovic, Z. (2007) Openfoam: A C++ library for complex physics simulations. International Workshop on Coupled Methods in Numerical Dynamics, IUC, Dubrovnik, Croatia. [16] Menter, FR. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal 32(8): 1598-1605, 1994. [17] Mason-Jones A., O’Doherty DM., Morris CE., O’Doherty T., Byrne CB., Prickett PW., et al. Non-dimensional scaling of tidal stream turbines. Energy; 44(1): 820-829, 2012. [18] Ammara, I., Leclerc, C. and Masson, C. A viscous three- dimensional differential/ actuator disk method for the aerodynamic analysis of wind farms. J. Solar Energy Eng., 124: 345–375, 2002. [19] Johansen, J. Sørensen, NN., Michelsen, JA. and Schreck, S. Detached-eddy simulation of flow around the NREL phase VI blade. Wind Energy, 5: 185–197, 2002. [20] Spalart, PR. Detached-eddy simulation. Annual review of fluid mechanics, 41: 181-202, 2009.

APPENDIX Table 1. Overview of experimental data for HATT

Source Measurement Location

Turbine Flow Condition Blade Clearance Shape #Blades Pitch D U0 (m/s) TI w0 (rad/s) l 1δFS (D)

2δSW (D) 3δLW (D)

Mason-Jones et al. [17]

Flume

Wortmann FX 63-137 3 3°, 6°, 9° 0.5m 1 2% 11 – 28 2.7 - 7 0.35 0.9 0.25

Morris et al. [13] Wortmann

FX 63-137 3 6° 0.5 1 2% 18 – 28 4.4 - 7 0.35 0.9 0.25

Tedds et al. [14] 0.5 – 1.5 2% 21.932 6.15 0.35 0.9 0.25


Ebdon et al. (unpublished) 1.5 1.8 –

2.1% 21.88 3.65 1.5 3.5 1.5

Ellis et al. (unpublished) Wave tank Modified Wortmann

FX 63-137 3 - 0.9m 1 - 8.89 4 1.67 4.5 2.22

Maganga et al. [3] Flume Tidal Hydaulic

Geneators Ltd design 3 0° 0.7m 0.5 – 1.5 8 – 25% 25.71 9 0.94, 1.57

2.04 - -

Eggar et al. [2]

Cleddau River

Tidal Hydaulic Geneators Ltd design 4 - 5.5m 0.9 10% 0.64 –

1.49 0.26 –

0.6 0.11 - - 1δFS : Blade tip clearance from free-surface. 2δSW : Blade tip clearance from the side-wall. 3δLW : Blade tip clearance from the bottom wall.





A REVISED ANALYSIS OF THE COMPRESSIBLE BOUNDARY LAYER AT HIGH REYNOLDS NUMBERS AND AT MACH 0.5

Tasmin Hossain, Julio Cesar Mendez North Carolina A&T State University

Greensboro, NC, U.S.A

Yang Gao, Frederick Ferguson North Carolina A&T State University

Greensboro, NC, U.S.A

ABSTRACT There are significant improvements in the development of

numerical methods for solving the system of Navier-Stokes Equations. However, many modern methods are designed for a specific purpose; either they are limited to solving compressible or incompressible flows, with either low or high Reynolds numbers. The Integro-Differential Scheme (IDS) may have the capability to surpass these limitations. The IDS is a recently developed numerical technique, which uses a unique combination of the integral and differential forms of the Navier-Stokes Equations. Unlike traditional control volume techniques, the IDS constructs the elementary control volume as a collection of spatial and temporal cells within a consistent averaging procedure. The test results from previous works [1, 6] including Quasi-1D flow, cavity flow and flow through an isolator demonstrate the accuracy with which the IDS captures the fluid physics. The aim of this study is to demonstrate the capability of IDS to solve very complex flow fields, such as the ones generated under the conditions of high Reynolds numbers, and under compressible flow field conditions. In doing so, a complex boundary layer flow over a flat plate is solved with the freestream Mach number of 0.5, and the Reynolds of one million that is based on the plate length. The research findings were compared to those obtained from the similarity compressible and incompressible solutions, independent DNS studies and with available experimental data. In all cases, the IDS predictions were commendable.

KEY WORDS: Navier-Stokes equations, incompressible flow, compressible flow, finite volume method, laminar boundary layer flow, Integro-Differential Scheme.

INTRODUCTION In 1905, Ludwig Prandtl created the hypothesis of a thin

layer that isolates the inviscid ideal flow dynamics from that of the nonslip surface. Prandtl postulated that for small viscosity, the resulting viscous force is negligible everywhere except for the thin boundary layer close to the solid boundaries where the no-slip condition must be satisfied, and within which, viscous effects dominate the fluid behavior. Prandtl also assisted in the

derivation of the boundary layer equations, which his student, Blasius, later solved analytically. The boundary layer concept is still considered one of the major break-throughs in the history of fluid mechanics. Today, it is a certainty that flows around a moving vehicle typically exhibit a thin layer along the solid body in which the relative flow velocity with respect to the body drops rapidly close to the solid walls. The fluid dynamics within the boundary layer are dominated by viscous effects, in contrast to the inviscid external flow. Depending on the body surface and the inviscid flow conditions, the resulting flow field within the boundary layer flow can exist in either a well behaved manner, forming a laminar flow field, or behave in a chaotic way, forming a turbulent flow field, or any combination thereof. Refer to Figure 1 where the boundary layer over an airfoil under conditions of interest to this study is illustrated.

Fig. 1: Illustration of the boundary layer over an airfoil

under compressible conditions at high Reynolds numbers

THE HIGH REYNOLDS COMPRESSIBLE FLAT PLATE CHALLENGE

The 1st International Workshop on High-Order CFD Methods was successfully convened in Nashville, TN, for 2 days in January 2012, Ref. [7]. This workshop was jointly supported by many agencies including the 50th Aerospace Sciences Meeting, the AIAA, the AFOSR and the DLR. The workshop generated a review article in which a set of intriguing CFD problems were identified. Solving these problems is now considered the major technical challenge facing the CFD community. Among these problems, was the Mach 0.5, high Reynolds number compressible flat plate boundary layer challenge. As part of the research effort described herein, a


detailed literature survey was conducted. The observations from this survey confirmed the 1st International Workshop on High-Order CFD Methods findings. Further, the literature survey conducted herein also confirmed that in the limited instances in which the high Reynolds number compressible flat plate boundary layer problem was solved, the solver needs to meet the following challenges:

i. High-order methods with highly clustered mesh ii. Resolving very steep velocity and temperature

gradients iii. Capability to capture transition from laminar to

turbulent iv. Compressible flow field v. Very thin boundary layers

Moreover, the success of the CFD methods was determined by their ability to compute the drag coefficient over the plate. The findings were compared to the results with that obtained with low-order methods.

Fig. 2: Expected fluid dynamic features in the

boundary layer under compressible conditions at high Reynolds numbers

As a test problem to evaluate the IDS capability, the Mach

0.5, ReL = 106, flow over the unit length flat plate at zero angle of attack is studied both analytically and numerically. The Reynolds number is computed based on the length of the plate (ReL = 106). Followed by the recommendation of Ref. [7], a constant ratio of specific heats of 1.4 and a Prandtl number of 0.72 is used. In addition, the dynamic viscosity must be computed using Sutherland's law.

The expectations of the fluid behavior within the developing boundary layer are illustrated in Figure 2. In addition, the numerical model developed herein is flexible enough to allow for the consideration of many types of boundary conditions around the numerical domain. In the case of a perturbed flow under zero pressure gradient, the instability of the laminar flow due to perturbations is studied. In the case of flows subjected to a pressure gradient or suction and blowing, laminar breakdown, separation bubbles and shock boundary layer interaction can also be reproduced at high Mach and Reynolds numbers. In this way, all the interesting phenomena can be captured, while the simplicity of the numerical framework is maintained. However, in the analysis described herein only the compressible CFD model is considered.

IDS – THE INTEGRAL DIFFERENTIAL SCHEME The laws of conservation of mass, momentum and energy

govern the physical characteristics of flow phenomena contained within the high Reynolds number compressible flat

plate boundary layer. In general, three approaches are followed when studying the boundary layer phenomena: experimental approach, computational or numerical approach, and to a limited extent, analytical or theoretical approach. However, irrespective of the solution approach, the underlying physics dictating the dynamics of the boundary layer flow are the conservation laws. These laws—the conservation of mass, momentum and energy laws—when coupled together form the Navier-Stokes system of equations. In mathematical terms, the Navier-Stokes equations represent an initial boundary value problem. Using the elementary results from boundary layer and inviscid theories, reliable initial and boundary conditions are derived and simulated.

In the research conducted herein, the numerical path, which has the advantage of providing detailed results over the experimental approach is employed. However, the existing experimental and theoretical results provided the much-needed guidance this effort required. Of greater importance to this study is IDS Navier-Stokes solver and the nature of the numerical computations methods used in the solver. The numerical schemes within the IDS solver were built with extensive physics-based considerations and have the following features:

i. The numerical scheme is based on the solution of the integral form of the Navier-Stokes equation. This approach focuses on the benefits of the traditional finite volume and finite difference schemes, and therefore guarantees the conservation properties throughout the domain by the first and the formulation simplicity by the latter.

ii. The Cartesian grid generation procedure is used to develop spatial and temporal control volumes upon which the integral form of the physical conservation laws are applied. As such, the scheme has the potential to satisfy the physical realities of fluid fluxes for both time and space.

iii. Accounting of the mass, momentums, and energy fluxes (both within the control volume and through its surfaces) is conducted with the aid of the mean value theorem, rather than the traditional extrapolation or interpolation of the node’s information from neighboring cells.

iv. The accuracy and efficiency of the scheme are demonstrated on flows with rectangular boundaries.

RESEARCH OBJECTIVES The research effort conducted for the report described

herein has a single goal. The goal is to numerically solve the Navier-Stokes equations accurately, thus capturing the detailed fluid dynamics within this high Reynolds number, Mach 0.5, boundary layer over the flat plate. In achieving this goal, the following objectives were set:

i. Objective 1: The computational IDS flow field solver that is very sensitive to the prescribed boundary conditions was analyzed to determine the best set of boundary conditions that describe the high Reynolds number compressible boundary layer over the flat plate.

ii. Objective 2: Grid Independence studies. The high Reynolds number compressible boundary layer over the flat


plate was analyzed on 4 - 6 sets of grids in an effort to determine the best grid resolution upon which the physical phenomena in the boundary layer will be studied.

iii. Objective 3: The physical phenomena within the high Reynolds compressible boundary layer were quantitatively compared to known results from the Blasius self-similar solutions to gain confidence in the CFD approach followed and to shed additional light on the findings. The expectation is the nature of the high Reynolds number compressible boundary layer is quite different from the Blasius boundary layer. However, the magnitudes of the primitive variables and the other flow field parameters are expected to reside within a reasonable range of those occupied by their respective counterparts in the Blasius cases.

iv. Objective 4: The findings from objective (iii) are used in combination with observations from previous analyses to identify the gaps in understanding the physics within the boundary layer. The stability of small perturbations within boundary layers, the laminar separation as predicted by semi-empirical theories, and even the quasi-steady and/or turbulent flow fields are still not fully understood.

v. Objective 5: Today, it is well established that only DNS-like computational approaches are capable of handling the complexities that are inherent in turbulent flows. Even though the IDS solver is not a DNS based code, a detailed investigation of its performance in the spatially and temporally developing flat plate boundary layer under increasing complexity will be numerically modeled and studied.

GOVERNING EQUATIONS The equations that govern fluid flows and the associated

heat transfer are the continuity, momentum and energy equations. These equations were independently constructed by Navier in 1827 and by Stokes in 1845. In this analysis, these equations are referred to as the Navier-Stokes equations. In this research, the integral forms of the Navier-Stokes equations, (1–3) are of paramount importance. The continuity, momentum and energy equations are listed as follows:

(1)

(2)

(3)

In Equations (1) the symbols, , represent the density, the volume of a control fluid element, and time, respectively. In addition, the symbols, , and , in equations (1) (2) and

(3) represent the fluid velocity, the surface of the control volume and the local heat transfer rate. In this research, fluid velocity and the surface element are described through the use of 2D vector quantities as follows:

(4)

(5)

(6)

In equations (2) and (3), the symbol, P, represents the pressure and the symbol, represents a symmetric tensor that defines the various components of the local viscous stresses. This symmetric tensor is described by the equation:

(7)

where the symbols of the six independent components, and are the local shear stress that were defined

in [4, 5] as follows:

(8)

(9)

(10)

The symbols, and , in equation (6) represent the components of the heat flux vector in the x- and y-directions, respectively. These components are defined by Fourier’s law, and expressed mathematically as,

(11)

The symbols, P and E, in equations (2) and (3) are defined as follows:

(12)

where R is the gas constant. The symbols, and k, represents the viscous and thermal properties of the fluid of interest. In this analysis, the viscosity of the fluid is evaluated using Sutherland’s law,

(13)

0v s

dv Vdstr r¶

+ =¶òòò òò

( ).

ˆv s

s s

Vdv V ds Vt

Pds ds

r r

t

¶+

¶

= - +

òòò òò

òò òò

.

ˆ. .v s

s s s

Edv EV dst

PV ds Vds qds

r r

t

¶+ =

¶

- + +

òòò òò

òò òò òò !

tv,,r

V sd q!

V u i v j= +

ds dy i dx j= +

x yq q i q j= +! ! !

,t̂

0ˆ 0

0 0 0

xx xy

yx yy

t t

t t t

é ùê ú

= ê úê úë û

, ,xx xy yxt t t yyt

( )2 . 23xx

uVx

t µ µ ¶= Ñ +¶

xy yxu vy x

t t µæ ö¶ ¶

= = +ç ÷¶ ¶è ø

( )2 . 23yy

vVy

t µ µ ¶= Ñ +¶

xq! yq!

;x yT Tq k q kx y¶ ¶

= - = -¶ ¶

! !

2 2;

2vu vP RT E C Tr +

= = +

µ

( )3/2 110 110T T T Tµ µ¥ ¥ ¥= + +


and are freestream values. In the case of 3D aerodynamic analysis, the Navier-Stokes

equations (1)–(3) defined above can be treated as a closed system of five equations relative to five unknowns. The unknowns are the following four primitive flow field variables:

. The immediate goal of this research is to develop an explicit method that solves the Navier-Stokes system of equations as defined herein.

THE BOUNDARY-LAYER EQUATIONS The Navier-Stokes equations described in equations (1 - 3)

can be simplified into two sets of boundary layer equations; one set of interest to compressible and the other of interest to incompressible flows. In this analysis, both sets of equations are of interest. These sets of equations were derived based on Prandtl’s assumptions. Prandtl postulated that (1) the boundary layer height is much, much less than the characteristic length of the plate, and (2) the ratio of the inertial to viscous forces has roughly the same order of magnitude within the boundary layer. Prandtl’s assumptions lead to the following ‘order of magnitudes’ relations: 𝑢 ≃ 𝑈, '(

')≃ *

+, '

,('-,

≃ *.,

, /01234567489:(8

=

𝑅𝑒+.,

+,, and .

+= >

?@1A. Using these relationships, the following

two sets of boundary layers equations were derived.

Compressible boundary layer equations The compressible boundary layer equations can be

expressed in the form,

(14)

(15)

(16)

(17)

where is the Prandtl number representing properties evaluated within the free stream. The compressible boundary conditions can now be written as,

(18)

Incompressible boundary layer equations In a similar manner, if the pressure gradient in the flow

direction, , is neglected the incompressible boundary layer is derived in the form

(19)

(20)

The boundary conditions for these equations are: y=0, u = v = 0, y= , and u= , assuming that the leading edge of the plate is x=0 and the plate is infinitely long. It can be observed that the assumption of neglecting the pressure gradient term,

, in the Blasius boundary layer equations leads to the v-component of the velocity vector having a steadily increasing value as it passes the boundary layer and travels deep into the freestream flow. This behavior does not support the physics of boundary layer flows, and the research effort conducted herein seeks to analyze the influence of the pressure gradient term.

SIMILARITY BOUNDARY LAYER SOLUTIONS In this analysis, both sets of similar boundary layer

equations, compressible and incompressible, are of interest to the analysis and both sets are utilized. The solution parameters from these analyses provided the foundational guidance the IDS solver requires. Once the IDS code is fully verified, detailed turbulence studies will be conducted. Incompressible boundary layer solution

Consider the system of incompressible equations illustrated in equations (14 - 18). This system of equations can be simplified further to an ordinary differential equation, when the self-similar transformation, , suggested by Blasius is used. With this transformation, The stream function,

, can be found from the continuity equation as , where is the dimensionless stream function. In a similar manner, using the definition of the stream function with respect to the Blasius transformation variable, , the velocity components, u and v, are derived in the following forms:

(21) and

(22)

The vorticity of the flow in the boundary layer as well as the shear force, , can also be derived from their respective

definitions; and , in the form of the

Blasius transforms as follows:

(23)

¥µ ¥T

[ ]u v Tr

( ) ( )0

u vx yr r¶ ¶

+ =¶ ¶

u v p uu vx y x y y

r r µæ ö¶ ¶ ¶ ¶ ¶

+ = - + ç ÷¶ ¶ ¶ ¶ ¶è ø

0py¶

=¶

2

2

Prh h p hu v ux y x y y

uy

µr r

µ

æ ö¶ ¶ ¶ ¶ ¶+ = + ç ÷¶ ¶ ¶ ¶ ¶è ø

æ ö¶+ ç ÷ç ÷¶è ø

Pr pc µ k¥ ¥ ¥=

( ) ( ) ( )( ) ( )

0,00,0 0; 0,0 0; 0

0; 0

w

y y

u v h h

u U h h¥=¥ =¥

= = - =

- = - =

p x¶ ¶

0u vx y¶ ¶

+ =¶ ¶

2

2υu v uu vx y y¶ ¶ ¶

+ =¶ ¶ ¶

¥ ¥U

0p x¶ ¶ =

y U vxh ¥=

f ( )vxU ff h= ¥

( )f h

h

' ( )u U f h¥=

1 ( ' )2vUv f f

x xf h¥¶

= - = -¶

xytu vy x

w ¶ ¶= -¶ ¶ xy

uy

t µ ¶=¶

( )2U x fw µ u h¢¢=


(24)

Finally, using the expressions for u and v from equations (21) and (22) can transform the boundary layer x-momentum equation relative to the non-dimensional stream function, into the Blasius differential equation and its respective boundary conditions:

(25)

(26)

Compressible Blasius boundary layer

In this analysis the compressible Blasius boundary layer equations (14 - 17) are evaluated for specified values of the specific enthalpy, h, at the wall. However, unlike the incompressible case, the density, ρ, viscosity, m, and thermal conductivity, κ, are not constants within the boundary layer.

In efforts to analytically solve the compressible boundary layer using the mathematical techniques used for the incompressible Blasius solution, an attempt is made to map the compressible boundary layer into the incompressible boundary layer. Refer to Figure &&.

Fig. 3: Mapping the compressible boundary layer to a

virtual incompressible counterpart Using this mapping concept, a compressible similarity solution emerges only if the following transformations in the flow field variables are imposed: , , , ,

, and . In addition, the pressure gradient, , in the flow direction must be neglected, and the value

of the enthalpy at the wall, , must be held constant. Under these assumptions, the compressible boundary layer becomes self-similar, and the variables of interest to the Howarth–Dorodnitsyn transformation become

(27)

Also imposed on the compressible similarity solution are the non-dimensional fluid parameters in the form,

(28)

where in this case, the density and viscosity were defined in terms of the non-dimensional enthalpy as follows:

. Finally, the compressible boundary layer equations are reduced to the following two equations,

(29)

where the symbols, and M, are the specific heat ratio and the Mach number, respectively. The boundary conditions can now be expressed in the form of the non-dimensional stream function as follows:

(30)

In general, equations (27 - 30) can only be solved once the density and viscosity variables, are specified. It

is of interest to note that under the appropriate assumptions, equations (29) are reduced to the incompressible momentum equation (20), and as such the solution for the non-dimensional stream function, f, can be used to solve the updated energy equation: . Further, when the non-dimensional stream function, , is used, an approximate solution for the self-similar compressible boundary layer temperature is generated in the form,

(31)

THE IDS SOLUTION TECHNIQUE The primitive variables associated with equations (1 - 13) are evaluated using the IDS procedure, which was described in Ref. [1-6], and the details are not repeated herein. In the developmental stages of this research and for illustration purposes, a typical flow field is represented by a rectangular domain and a typical elementary fluid element represented by a rectangular prism; refer to Figure 4. Figure 4 represents the Integro-Differential Model (IDM) as it is applied to the computational solution to the NSE (1-3). In general, the IDS solution of a given fluid dynamic problem is built on an interconnecting set of spatial and temporal cells. In the Cartesian system of coordinates, a typical fluid cell is nothing more than a carefully chosen elementary rectangular prism, defined by the dimension; dx, dy and dz. It is the application of a specified fluid cell in relationship to the NSE equations that determines whether it becomes a spatial cell or a temporal control volume.

( )2xy U x ft µ u h¢¢=

( )f h

0ff f¢¢ ¢¢¢+ =

( 0) 0; ( 0) 0( ) 1

f ff

h hh¢= = = =®¥ =

2x L x® y Ly® u u® 1v L v-®h h® r r® µ µ®

p xD D

wh

( )0

,2 2

yU dy fv x v Ux

r yh hr¥ ¥ ¥

= =ò

( ) , , ,ww

h hh hh h

r µh r µr µ¥ ¥ ¥

= = = =! ! ! !

1 2 3,h hr µ-= =! !! !

( )

( ) ( ) 2 2

2 0

1 0

f ff

h Prfh Pr M f

rµ

rµ g rµ

¢¢¢ ¢¢+ =

¢¢ ¢ ¢¢+ + - =

! !

! !! !! !

g

( 0) 0; ( 0) 0

( 0) 0

( ) 1; ( ) 1

f f

h h

f h

h h

h

h h

¢= = = =

= - =

¢ ® ¥ = ®¥ =

! !

!

( ) ( ),h hr µ! !! !

0Pr f h h¢ ¢¢+ =! !

f

( ) ( )0 0 0

Pr Prw

w

T Tf d f d

T T

hh h

¥- ¢¢ ¢¢=- ò ò


Fig. 4: Spatial cell with notation at surfaces nodes

A given cell is defined locally by six independent surfaces,

and each surface defined by four points or nodes, with the set of four nodes lying in a given plane. Additionally, plus and minus notations are used to define the unit normal, n, with respect to each surface. Next, each surface of each cell is defined by four nodes. Each term in the Navier-Stokes Equations (1 -13) is applied systematically to each cell, and thus they are called spatial cells. The mean value theorem is invoked, and a set of algebraic equations representing the rate of change of mass, momentum, and energy associated with each spatial cell is derived. However, the rates of change of the time-fluxes are not associated with any grid point, but with the spatial cell. Analogous to the spatial cells, the concept of a temporal cell is also introduced. The temporal cells are defined as rectangular prisms formed from the center points of the eight neighboring spatial cells. Finally, the concept of a computational control volume is defined, as a collection of eight spatial cells and one integrated temporal cell. An IDS representation of the computational control volume is illustrated in Figure 5.

Fig. 5: IDS computational control volume

It is of interest to note that inviscid fluxes are computed with the local information of the spatial cells. However, two additional and adjacent surfaces in each directions are needed for evaluating the viscous terms.

Finally, the time marching technique is based on the Taylor’s expansions series, and it is shown in equation (11).

(32)

This implementation involves a unique technique to compute the right-hand side terms from equations (11) where an

averaging procedure is used to compute the solution vector and the timer derivative; interested readers refer to [5].

THE IDS FLAT PLATE ANALYSIS The case of interest considers subsonic, laminar flow over

a flat plate with freestream Mach number 𝑀C = 0.5, and the Reynolds number, 𝑅𝑒+ = 1 × 106. The Prandtl number is set to Pr = 0.72. As the Mach number is greater than 0.3, the compressibility effect needs to be considered. The challenges behind this problem lie in the fact that, the compressibility effect coupled with the high Reynolds number results in a turbulent behavior inside the boundary layer.

Fig. 6: Computational domain for the flat plate

boundary layer The flow field along with the boundary conditions of

interest to this study is illustrated in Figure 6. The computational domain has two other length scales LH and LV, as shown in Figure 2. In particular the flow field physics within the boundary layer over the plate is studied is under adiabatic conditions. As part of this analysis, the influence of the length scales parameters, LH and LV, on the numerical analysis are studied. In addition, the ranges of these parameters are chosen to be large enough values so as not to affect by flow physics within the boundary layer.

Our computational domain is a rectangular one with a leading edge (LE) gap of 𝐿𝑒 = 0.5 and plate length 𝐿 = 1.0 in Fig.4. The height of the domain is taken as, ℎ = 0.01875 to accomodate for the sudden changes inside the very thin boundary layer, the theoretical non-dimensional boundary layer height being 𝛿 = 0.00491. The height and length are in non-dimensional form, as IDS solves the non-dimensional form of the NS equations.

IDS GRID INDEPENDENCE STUDY As part of the grid independence study, the high Reynolds

compressible flat plate problem was numerically solved by the IDS procedure on four sets of rectangular grids. The dimension of the flow domain, (LH by LV), was set to (1.5 by 0.01875), which in turn is represented by the grid system M by N. Also in the grid independence study the values of M and N were set equal to each other. The specific set of grids used in this analysis is tabulated in Table 1. In each case, the solution set was iterated until the criteria for the error norm reached a value of less than 10-7. In accordance with the IDS computational procedure the error norm is defined as the maximum change in mass, momentum and energy fluxes across the domain during each level of iteration. In addition, at the end of each run for a given grid set, the v-velocity and skin friction coefficient among with

( ) ( ) ( ) tt+ t ti, j i, j

i, j

d PVPV = PV + t

dtD æ ö

Dç ÷ç ÷è ø


many other fluid dynamic parameters, along the plate and within the domain were recorded and analyzed.

Table 1: IDS Boundary Layer Grids

As part of this report, the distribution of the variation in the v-velocity and skin friction coefficient along the plate are plotted in Figures 7 and 8. Figure 7 showed that for coarse grids the IDS predicted a velocity profile that is closely aligned with the Blasius profile derived from the similarity solution procedure. However, for fine grids the solution jumped to a profile that closely matches that of a typical turbulent boundary layer. The turbulent like feature of the boundary layer was further confirmed by the observations revealed in Figure 8. Figure 8 shows that for fine grids the skin friction coefficient closely aligns with the well-established turbulent quantities. It may be inferred from the data provided that the flow is transitioning from a laminar to a turbulent profile. However, a more detailed study is required before this fact is confirmed. Finally, a careful observation of the independence results shows that an acceptable grid in the flat problem is one defined by a value of N = M = 4001.

Fig.7: Dimensionless u-Velocity profile at first quarter of the plate compared to the Blasius

profile.

IDS FLAT PLATE SOLUTION Using the converged grid profile of 4001 by 4001, the IDS

primitive variables within the domain are studied. The contour plots of the non-dimensional density, velocity components, u and v, and the temperature are shown in Figures 9 – 12. As expected, the flow field domain is primarily influenced by the

leading edge of the plate, and because the flow field is subsonic, this influence is distributed somewhat evenly in the x positive and negative directions as observed in Figures 9 and 11. Moreover, as confirmed by the four contour plots, the subsonic nature of the flow field at the specified Mach number of 0.5 purposefully allows for the sonic disturbances to travel strongly in the y-direction. Observing the plots in Figures 9 – 12 one cannot derive much information about the details within the boundary layer. As such, a detailed boundary layer analysis will be conducted in the next section. However, the flow field variables outside the boundary layer and away from the zone of sonic disturbances experience only minor changes, and as such, no further studies will be conducted in these regions. In addition, the leading edge of the plate, defined by the location x = 0.5 turns out to be the most sensitive and complex region within the domain, as such further study is warranted.

Fig.8: Variation of Skin Friction Coefficient

Fig. 9: Density contour for 2501 X 2501 Grid

Fig. 10: u Velocity contour for 2501 X 2501

Grid

Grid Size No. of Points in the Boundary

Layer Behavior

1501 X 1501 32 Laminar

2001 X 2001 42 Laminar

2501 X 2501 53 Turbulent

4001 X 4001 85 Turbulent


Fig. 11: v Velocity contour for 2501 X 2501

Grid

Fig. 12: Temperature contour for 2501 X 2501

Grid

IDS QUASI-UNSTEADY FLOW FIELD OBSERVATION In the process of simulating the flow field within the high

Reynolds compressible boundary layer, evidence of a quasi-unsteady field was observed. The observation was further probed as part of the so-called IDS ‘quasi-unsteady’ studies. Using the converged solution as a restart file, IDS solution procedure was allowed to continue for 55K, 95K and 155K cycles, respectively. During each cycle the behavior of the error norm was recorded and plotted in Figures 13 and 14. Figure 13 shows both the transient and the unsteady nature of the IDS high Reynolds compressible boundary layer. Further, Figure 13 demonstrates that in each case the transient behavior of the IDS flow field error norm was identical. Also, it showes that in each case, the unsteady behavior appeares at approximately the same time with approximately the same frequency profile. A close-up look at this fact is observed in Figure 14. The data provided in Figures 13 and 14 tends to support the fact that the flow field is unsteady.

Fig. 13: The transient and unsteady behavior

of the error during post convergence tests

Fig. 14: A close-up of the unsteady behavior of the error during post convergence tests

UNSTEADY BOUNDARY LAYER PROFILE IDS evaluation was conducted with efforts geared towards

observing the unsteady boundary layer profiles. In these tests all the primitive flow field variables were observed; however, herein only the v-velocity component and the pressure profiles are plotted. In Figures 15a and 15b, the profiles of the v-velocity component are plotted at the leading and trailing edges. In a similar manner, in Figures 16a and 16b the profiles of the pressure distribution at the leading and trailing edges are plotted. A close-up observation of these plots indicated the flow field within the vicinity of the wall the flow is steady; however, away from the wall, in the region where turbulence effects are known to be dominant, the flow is clearly unsteady. The data illustrated in Figures 16a and 16b support this fact.


Fig. 15a: Illustration of unsteady periodic behavior of the

v-velocity profile at the leading edge

Fig. 15b: Illustration of unsteady periodic behavior of the

v-velocity profile at the trailing edge

Fig. 16a: Illustration of unsteady periodic behavior of the

pressure profile at the leading edge

Fig. 16b: Illustration of unsteady periodic behavior of the

pressure profile at the leading edge

Fig. 17: Illustration of behavior of the temperature profile

at different locations for 95k run

Fig. 18: Illustration of behavior of the pressure profile at

different location for 95k run


When an alternative observation is taken, i.e., by comparing the degree of variation from one location to the next for varying times after convergence, that snap shots indicate that the magnitude of the changes in the primitive are quite small. However, the profiles seems to repeat themselves. The temperature and pressure plots illustrated in Figures 17 and 18 respectively supports this conclusion.

LAMINAR AND TURBULENT PROFILES Capturing the onset of turbulence is a very complex matter.

For turbulent flow over the plate, at first the boundary layer remains laminar. However, as it moves farther way from the leading edge, it tends to get more rapidly growing and thicker in nature due to the fluctuations. From the literature review it is confirmed that, the turbulent boundary layer is fuller, thicker and the profile has a greater slope than the laminar one. Now, if the IDS prediction of velocity profiles can be plotted in vector form, they reveal somewhat similar findings as stated before.

(a) Laminar Profile

(b) Turbulent Profile

Fig. 19: IDS prediction of laminar and turbulent velocity profiles

Figure 19 shows the velocity profiles taken (a) at the beginning of the flow and (b) at halfway of the plate when the flow is fully developed. These profiles clearly indicate the differences between laminar and turbulent profile. Also, the thickness of turbulent boundary layer as depicted by IDS is greater than the laminar one obtained close to the leading edge.

TURBULENT BOUNDARY LAYER PHYSICS The velocity profile for the turbulent flow over a flat plate

consists of a profile similar to the one illustrated in Figure 20. As noted in Figure 19, the typical turbulent u-velocity profile consists of a viscous sublayer, a buffer layer, and an inertial sublayer and a defect layer. More importantly, the velocity profile is mainly dictated by the non-dimensional variables, uτ,

y+ and u+, where these variables are defined as uτ = (τw /ρ)1/2, y+ = uτy/v, and u+ = u/ uτ, respectively. In addition, noted in Figure 15 are the defining patterns associated with each layer. For example, very close to the wall, defined by the values of y+ in the range of 0 to 7, constitutes the viscous sublayer. In this region, typically the turbulent features of the flow dampen, as the viscous sublayer is dominated by the effects the wall shear force, τw. In this region, the velocity profile is linear and defined by the expression, u+ = y+. The inertial layer, namely the region in which the turbulent effects are most dominant, the velocity profile is governed by the logarithmic relationship: u+= aln(y+) + b, where the coefficients, a & b, are defined by their respective turbulent correlations. In this region, the y+ values range from approximately 30 to 1000 or more. The defect layer is highly nonlinear and is dictated by the y+ values in the range 1000 and greater. The buffer layer is found between the viscous sublayer and the inertial layer, typically with y+ values ranging from 7 to 30.

Fig. 20: The typical u-velocity profile in turbulent

wall flows Having observed turbulent like behavior in the IDS

boundary layer prediction within the high Reynolds number compressible boundary layer, the u-velocity profile at the center line of the plate was transformed using the turbulent variables described in the preceding paragraph of this paper. The resulting IDS u-velocity plot is illustrated in Figure 21, and it is also compared to previous efforts conducted by Ref [8] and [9]. The PIV experimental result from Ref [11] is also plotted to showcase the highly compressible region and also to compare the IDS results with an experimental one for the viscous sublayer, as this layer is independent of Mach number. The results clearly showed that the IDS prediction is 100% on target in the viscous sublayer but a little off in the inertial layer. However, these results compare remarkably well in comparison to the efforts conducted in Ref [8-9].


Fig. 21: IDS turbulent boundary layer

CONCLUSION The aim of this study was to demonstrate the capability of

IDS Navier-Stokes solver in solving very complex flow fields, such as the ones generated under the conditions of high Reynolds numbers, and under highly compressible flow field conditions. To date, there is no true solution available to the flat plate problem of interest. Very few numerical solutions have been found in the open literature, each of which differs from the others. This led the study conducted herein to take on a completely qualitative nature. For the analysis described herein, the IDS solver was successfully used to solve the full set of Navier-Stokes equations, when the freestream Reynolds number was set to 1 million and the Mach number was set to a compressible flow value of 0.5. The aim of this project was achieved.

From the obtained IDS results, it can be concluded that the IDS was able to predict the flow field physics remarkably well. They are in very good agreement with the physical expectations as depicted in literature. IDS was able to predict the turbulence flow field without any form of turbulence models imposed within the scheme. A fully developed boundary layer was obtained as well as the subsonic flow features were visible due to the leading edge disturbances occurring within the flow. As part of the IDS solution validation process, grid independence evaluations and flow field investigations were conducted. The findings demonstrated that the IDS delivered turbulence-like solution; i.e., quasi-unsteady behavior, for finer grids. Moreover, this study concludes that there is strong evidence to support the fact that the high Reynolds number compressible boundary layer is either transitional or fully turbulent. To fully confirm the nature of the boundary layer, further IDS evaluations are planned.

REFERENCES [1] Elamin, G. A., “The Integral-differential Scheme (IDS): A New CFD Solver for the System of the Navier-Stokes Equations with Applications,” Ph.D. thesis, North Carolina Agricultural and Technical State University, 2008. [2] Mendez, J., Dodoo-Amoo, D., Dhanasar, M., and Ferguson, F., “Physics Based Validation Of An Improved Numerical Technique For Solving Thermal Fluid Related Problems,” 2017. [3] Ferguson, F., Mendez, J., Dodoo-Amoo, D., and Dhanasar, M., “The Performance Evaluation of an Improved Finite Volume Method that Solves the Fluid Dynamic Equations,” 2018 AIAA Aerospace Sciences Meeting, 2018, p. 0834. [4] Ferguson, F., Mendez, J., and Dodoo-Amoo, D., “Evaluating the Hypersonic Leading-Edge Phenomena at High Reynolds and Mach Numbers,” Recent Trends in Computational Science and Engineering, edited by S. Celebi, InTech, Rijeka, 2018, Chap. 4. [5] Frederick Ferguson, Mookesh Dhanasar, Jovan Brown, and Isaiah M. Blankson, ‘A Finite Volume Model For Simulating the Unsteady Navier-Stokes Equations Under Space Time Conservations’, AIAA-2013-2859, 43rd AIAA Fluid Dynamics Conference and Exhibit, 24–27 June 2013, San Diego, California [6] Frederick Ferguson, Nastassja Dasque and Mookesh Dhanasar, Validation and Verification of Navier-Stokes Solutions Obtained from the Applications of the Integro-Differential Scheme, AIAA-2011-1300, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, Jan. 4-7, 2011. [7] 1st International Workshop on High-Order CFD Methods January 7-8, 2012. Nashville, Tennessee, USA. [8] Jonathan Poggie, “Compressible Turbulent Boundary Layer Simulations: Resolution Effects and Turbulence Modeling”, Air Force Research Laboratory, Wright-Patterson AFB, Ohio. [9] Rai, M. M., Gatski, T. B., and Erlebacher, G., “Direct Simulation of Spatially Evolving Compressible Turbulent Boundary Layers,” AIAA Paper 95-583. [10] Frank M. White (1991), Viscous Fluid Flow. [11] Lee, H., Kim, Y. J., Byun, Y.H., Park, S.H., “Mach 3 Boundary Layer Measurement over a Flat Plate Using the PIV and IR Thermography Techniques”, 55th AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, AIAA 2017-0523.





AN OPEN CYCLE THERMODYNAMIC SIMULATION QUANTIFYING BENEFITS OF TURBOCOMPOUNDING IN DUAL FUEL COMBUSTION

Adithya Baburaj1,2, Abhinandhan Narayanan1,2, Sundar Rajan Krishnan1, Kalyan Kumar Srinivasan1, *

1The University of Alabama Tuscaloosa, Alabama, USA

2SASTRA Deemed-to-be University Thanjavur, Tamil Nadu, India

ABSTRACT A zero-dimensional, single zone, open cycle simulation of a

compression ignition, single cylinder research engine (SCRE) operating on diesel pilot-ignited methane dual fuel combustion was developed to characterize the benefits of turbocompounding. Dual fuel combustion was simulated for three different diesel injection timings or start of injection (SOI): 340 crank angle degrees (CAD) after gas exchange top dead center (aGETDC), 350 CAD aGETDC and 355 CAD aGETDC, for an engine load of 3.3 bar brake mean effective pressure (BMEP) at a methane energy substitution of 80 percent, an engine speed of 1500 RPM and a boost pressure of 1.48 bar. Results for in-cylinder pressure, trapped mass at intake valve closure (IVC) and apparent heat release rate (AHRR) were predicted within 3% error of the corresponding experimentally obtained values. The net indicated fuel conversion efficiency (IFCE) was predicted to within 9-12% of the experimental values. Subsequently, the simulation was used to predict the theoretical maximum performance improvements possible using turbocompounding, with exhaust manifold pressure being explicitly set equal to the in-cylinder pressure at exhaust valve opening (EVO) (obtained from the model). Exploratory studies were performed to determine the efficiencies at which the turbocompounding turbine must operate such that the exhaust gas, after expansion, has sufficient enthalpy to run a turbocharger for delivering the required boost pressure. At 355 CAD SOI, the net IFCE increased from a baseline (non-turbocompounding) value of 25% to a maximum value of 37% at assumed turbocharger and turbocompounding isentropic efficiencies of 80%.

AHRR Apparent Heat Release Rate RPM Revolutions per minute PM Particulate Matter IFCE Indicated Fuel Conversion Efficiency aGETDC After Gas Exchange Top Dead Center SOI Start of Injection TDC Top Dead Center

KEYWORDS: Open cycle simulation, Dual fuel combustion, Apparent heat release rate, Net indicated fuel conversion efficiency, Turbocompounding,

INTRODUCTION The challenges of stringent emission regulations and the

availability of limited petroleum supplies can be overcome only by developing and applying appropriate technologies to the entire vehicle system. However, as the prime mover of choice in most modern vehicles, the characteristics of an internal combustion engine play a major role in striking a balance between the pollutants generated, power density and fuel economy. To achieve simultaneous reduction of NOx and PM emissions, dual fuel combustion, using a variety of fuels including natural gas and propane with diesel as an ignition source, has been investigated. Natural gas, due to its excellent knock resistance, facilitates higher compression ratios and in turn better thermal efficiencies. With the US Department of Energy’s SuperTruck-II program, there has been a paradigm shift in the efficiency targets for the heavy-duty engine industry, with 55% brake fuel conversion efficiency being considered an achievable target [1]. It is highly likely that such a high efficiency target cannot be achieved by the sole use of in-cylinder combustion strategies like rapid combustion, high pressure fuel injection (>3,000 bar), Atkinson cycle adoption and other low temperature combustion strategies like gasoline compression ignition (GCI) [2], homogeneous charge compression ignition (HCCI) [3] and dual fuel low temperature combustion [4]. Also, for dual fuel combustion, it is desirable to improve the fuel conversion efficiency, especially at part loads. One approach that is being considered for augmenting heavy-

ABBREVIATIONS BMEP Brake mean-effective pressure IVC Inlet valve closure EVC Exhaust valve closure EVO Exhaust valve opening IVO Inlet valve opening CAD Crank angle degree


duty engine efficiency is exhaust waste energy recovery. Existing waste energy recovery methods such as organic Rankine cycles have high cost per unit power output [1]. Therefore, direct exhaust energy recovery approaches that may be less expensive are attractive. The present work examines one such approach (turbocompounding) for diesel-methane dual fuel combustion in a single-cylinder heavy-duty diesel engine. Turbocompounding [5] in IC engines has been in existence since at least 1944, when it was used in aircraft engines. With turbocompounding, a turbine was used to expand exhaust gases and extract work from it to drive the aircraft propeller.

From the earliest days of IC engine development, modeling the overall engine operation to understand and to improve engine technology has been a constant desire and aim among engine researchers. Engine models have been explored since the late 1800s; however, they involved several oversimplifying assumptions, and therefore, were less accurate [6]. These models were called the ideal air standard cycles and could provide trendwise insights into engine operation, which was their primary advantage. Some improvements were made in the 1930s, by incorporating more realistic thermodynamic properties for mixtures; nevertheless, improvements in model accuracy were quite modest, as other simplifying assumptions were still present. The advent of thermodynamic simulations in the 1960s was a great improvement compared to previously available models. The significant time gap from the 1930s to the 1960s can be attributed primarily to the unavailability of sufficient computing capability, and partly to the limited understanding of the inherent physical processes [6]. By the 1970s, full-fledged thermodynamic simulations which predicted engine out emissions were being developed. Patterson and van Wylen [7] developed one of the first thermodynamic models for a spark ignition engine in 1964; a two-zone model comprising a burned and unburned zone was built by considering mixture composition, combustion chamber shape, manifold pressure and temperature, and exhaust pressure. This simulation did not account for heat transfer between the unburned and burned regions, but it was a significant development at that time. McAulay et al. [8] and Krieger and Borman [9] developed a computer simulation for compression ignition engines in 1965. Their primary aim was to reduce engine design and development time. The next generation of models built upon the thermodynamic framework by considering chemical kinetics. The problem of nitric oxide formation was addressed by using the Zeldovich reaction mechanism. One of the first models of this type was built by Lavoie et al. [10] and was verified with experimental measurements of nitric oxide emissions. Heywood et al. [11] developed one of the first models with three zones for the combustion process for accurate prediction of nitric oxide. The burned zone was divided into an adiabatic core and a boundary layer zone, in addition to a separate unburned zone. It was observed that the presence of an adiabatic core was quite important in predicting the nitric oxide emissions, as they were highly dependent on temperature.

An open-cycle thermodynamic simulation has been developed in the present work, wherein the gas exchange processes have been modeled using one-dimensional isentropic flow equations to capture the crank angle-resolved mass inflow and outflow from the engine. Modeling the intake and exhaust processes gives an idea about the mass trapped in the cylinder at the end of the intake stroke and the amount of residual exhaust gas present in the cylinder at the end of the exhaust process, both of which are quite important for characterizing and improving engine performance. The first law of thermodynamics, serves as the starting point for such models. The idea is to simplify the energy equation in such a way that either pressure or temperature is the only unknown variable in the equation, and the other variable can be determined using the ideal gas equation of state. The present simulation was performed for different loads and validated against experimentally obtained trapped mass at IVC and in-cylinder pressure histories. In addition, investigations pertaining to improving the fuel conversion efficiency by recovering some part of the exhaust exergy (available energy) using a hypothetical turbocompounding setup were completed.

ENGINE DESCRIPTION The engine considered for the present study is a single

cylinder, four-stroke, compression ignition engine. Engine specifications are provided in Table 1. Dual fuel combustion experiments were conducted for various operating conditions, and mass flow rates and in-cylinder pressure histories were recorded. In-cylinder pressure was measured using a Kistler model 6052C pressure sensor and a Kistler 5010B type charge amplifier. Pilot diesel and methane mass flow rates through the valves were measured with Emerson Micro Motion coriolis mass flow meters [12]. The methane PES was 0.8 under these conditions Table 1. Single Cylinder Research Engine Specifications

[12] PARAMETER SPECIFICATIONS Bore x Stroke (mm x mm) 128 x 142 Connecting rod length (mm) 228 Compression ratio 17.1:1 Displaced volume (cc) 1827 IVO (CAD) 3.6 IVC (CAD) 174.6 EVO (CAD) 503 EVC (CAD) 712 Maximum speed (RPM) 1900

MODELING APPROACH The simulation has a few simplifying assumptions which are

stated below: 1. The gas-exchange processes have been modeled using

one-dimensional isentropic flow equations. 2. Working fluid (air+methane) is modeled as an ideal gas.


3. Pressures downstream of the intake valve and upstream of the exhaust valve are assumed to be equal to instantaneous values of the in-cylinder pressure, which is uniform throughout the cylinder.

4. Variation of specific heat ratio with temperature can be modeled using the Zucrow and Hoffmann [13] equation.

Engine kinematics are modeled using Equations (1) and

(2). The volume of the cylinder as a function of crank angle can be represented as:

𝑉(𝜃) =𝑉)*+,𝑟 − 1 +

𝑉)*+,2

11 + 𝑅 − cos𝜃 –7𝑅8 − 𝑠𝑖𝑛8𝜃<(1)

In Equation (1), 𝑉)*+, is the displaced volume, 𝑟 is the compression ratio, 𝑅 is the ratio of connecting rod length to crank radius, and θ is the crank angle.

The surface area of the combustion chamber as a function of crank angle can be represented as:

𝐴(𝜃) = 𝜋𝐵8

2 + 𝜋𝐵𝑆211 + 𝑅 − cos𝜃 –7𝑅8 − 𝑠𝑖𝑛8𝜃<(2)

In Equation (2), 𝐵 refers to the bore of the engine, and 𝑆 refers to the stroke length.

The amount of mass flow into/out of the cylinder per unit time can be represented as [14]:

�̇�𝒂𝒄𝒕𝒖𝒂𝒍 =𝑪𝒅𝒑𝟎𝑨𝒗𝜸7𝜸𝑹𝑻𝟎

Q 𝒑𝒑𝟎R𝟏𝜸 T 𝟐

𝜸V𝟏W𝟏 − Q 𝒑

𝒑𝟎R𝜸X𝟏𝜸 Y(𝟑)

In Equation (3), the subscript 0 represents upstream conditions (inlet manifold for intake and in-cylinder for exhaust), and downstream conditions (in-cylinder for intake and exhaust manifold for exhaust) are represented by 𝑝 and 𝑇(downstream pressure and temperature). 𝐶) is the coefficient of discharge, Av is the valve open area,𝛾 is the specific heat ratio. The values of 𝐶) for both the intake and the exhaust valve are chosen such that the difference between the predicted trapped mass and the experimental value is minimized. There are two intake and exhaust valves present in the cylinder. The 𝐶) value for each intake valve is taken as 0.45 and 0.5 for each exhaust valve.

The basic energy equation that governs the engine processes can be derived by applying the first law of thermodynamics to a system which consists of the charge inside the cylinder as follows: 𝑑𝑇𝑑𝜃 =

𝑚b𝑄def𝑚𝐶f

𝑑𝑥h𝑑𝜃 −

ℎ𝐴(𝑇 − 𝑇j)𝑚𝐶f𝜔

−(𝛾 − 1)𝑇

𝑉𝑑𝑉𝑑𝜃

+�̇�*l𝑇𝑚𝜔

m𝛾𝑇n𝑇 − 1o −

�̇�pqr𝑇𝑚𝜔

(𝛾 − 1)(4)

In Equation (4),𝑚b is the amount of fuel added per cycle,

𝑄def is the lower heating value of the fuel, )tu)v

is the rate of the mass burnt fraction inside the cylinder during combustion, m is the instantaneous mass inside the cylinder, 𝜔 is the angular speed of the engine, ℎ is the heat transfer coefficient, 𝐴 is the surface area of the cylinder, cv is the specific heat at constant volume, 𝑇 is the instantaneous in-cylinder temperature, 𝑇j is the wall temperature, )w

)v is the rate of change of volume with respect to

CAD, �̇�*l is the rate at which mass enters the cylinder, 𝑇n is the inlet air temperature, γ is the specific heat ratio, �̇�pqris the rate at which mass leaves the cylinder.

In Equation (4), for the intake process, the first term on the right-hand side is zero (𝑚b = 0), and so is the last term on the right-hand side (�̇�pqr = 0) (because the present engine did not have any valve overlap). For the exhaust and expansion (after EVO) processes, the first term on the right-hand side is zero (𝑚b = 0) and so is the second last term on the right-hand side (�̇�*l = 0). For compression and expansion (from IVC to EVO) processes, every term on the right-hand side is zero, except the second and third terms. This is because no mass enters or leaves the system. 𝑄def = 𝑚)𝑄) + 𝑚y𝑄y(5)

In Equation (5),d and g refer to diesel and natural gas respectively, 𝑚 and 𝑄 refer to the amount of fuel and the lower heat value of the fuel respectively.

COMBUSTION MODELING Combustion models form an integral part of every engine

cycle simulation. In this work, a zero-dimensional, single zone model has been developed to characterize the combustion process. Single zone as well as two-zone models can be used in two distinct ways as explained in Figure 1 (from Ref. [15]).

In this work, the first method is followed, also the start and end of combustion are determined using experimental AHRR histories. A double Wiebe function developed by Miyamoto et al. [16], wherein the two Wiebe functions are used to express the heat release rates, is fit to the experimentally obtained heat release rate histories using a non-linear least-squares fit.

The values obtained as outputs from the fit are used to determine the in-cylinder pressure as a function of crank angle during the combustion process. Miyamoto expressed the heat release rates rather than the mass burnt fraction in the form of a double-Wiebe function as expressed in Equation (6). )}~)v

= 6.9 }�v�(𝑚l + 1) m

vVv�v�o��exp �−6.9 mvVv�

v�o(��)

�

+ 6.9 }�v�(𝑚) + 1) Q

vVv�v�

R��exp �−6.9 QvVv�

v�R(��)

�(6)


Figure 1. Directions followed by single zone models [15].

In Equation (6), the subscripts p and d refer to premixed and

diffusive phases of combustion. 𝜃, and 𝜃) are the durations of energy release, 𝑄, and 𝑄) are the amounts of energy release, 𝑚, and 𝑚) are shape factors. Thus, this double-Wiebe function contains six adjustable parameters, each of which can be determined by the least-squares fit, such that the model function best fits the experimental data.

HEAT TRANSFER MODELING Heat transfer due to convection between in-cylinder

contents and the walls of the cylinder has been modeled using Woschni’s correlation [17], which can be expressed as follows:

ℎ = 3.26(𝐵)Vn.8(𝑃)n.�(𝑇)n.��(𝜔)n.�(7) In Equation (7), ℎ is the instantaneous spatially averaged heat transfer coefficient in W/(m2K), 𝐵 is the bore in m, 𝑃 is the instantaneous in-cylinder pressure in kPa, 𝑇 is the instantaneous in-cylinder temperature in K and 𝜔 is the average cylinder gas velocity in m/s. 𝜔 can be determined by:

𝜔 = �𝐶�𝑆, + 𝐶8𝑉)𝑇�𝑝�𝑉�

(𝑃 − 𝑃�)�(8)

In Equation (8) 𝑉) isthe displaced volume in m3,𝑆,is the

mean piston velocity in m/s. 𝑝� ,𝑉� and 𝑇� are the reference pressure, volume and temperature.𝑃�is the motoring pressure. 𝐶�and𝐶8are constants; their values depend on the process. For gas exchange processes: 𝐶� = 6.18 and 𝐶8 = 0.For compression process: 𝐶� = 2.28 and 𝐶8 = 0.For combustion andexpansion:𝐶� = 2.28and𝐶8 = 3.24 ∗ 10V�.

SPECIFIC HEAT RATIO MODELING The specific heat ratio (𝛾) is one of the most important

parameters, as it directly influences the net apparent heat release rate Q�}~

)vR as follows:

𝑑𝑄l𝑑𝜃 =

𝜕𝑄�e𝑑𝜃 −

𝜕𝑄er𝑑𝜃 =

1𝛾 − 1𝑉

𝑑𝑃𝑑𝜃 +

𝛾𝛾 − 1𝑃

𝑑𝑉𝑑𝜃 (9)

�}��)vis the chemical energy release rate, and �}��

)v is the rate of

energy transfer due to heat transfer. Hence, an accurate determination of 𝛾 is necessary. Assuming a constant value for γ leads to errors. In this work, the specific heat ratio has been modeled using the Zucrow and Hoffman equation [13]. γ = 𝐶� + 𝐶8𝑇 + 𝐶�𝑇8(10) In Equation (10) 𝐶1, 𝐶2, and 𝐶3 are determined by solving the equation at points where the γ value is known, i.e. γ = 1.389 at 500 K, γ = 1.336 at 1000K and γ = 1.298 at 2000K [18].

TURBOCOMPOUNDING SETUP There is significant loss of the potential to perform useful

work (i.e., exergy) during the exhaust blowdown process in any four-stroke engine. In performing the turbocompounding simulations, we hypothetically eliminate the pressure difference between the in-cylinder conditions and the exhaust manifold conditions by explicitly setting the exhaust manifold pressure to be the same as the in-cylinder pressure at EVO. Consequently, the exhaust blowdown process can be eliminated, and the high-pressure exhaust can be used to drive a turbocompounding turbine to improve net IFCEs. One key requirement for this turbocompounding setup is to ensure that the gas exiting the turbocompounding turbine has sufficient enthalpy to run a standard turbocharger that is required to deliver the required boost pressure of 1.48 bar. The turbocompounding setup used in the present work is shown in Figure 2.

The assumption of a relatively high (constant) pressure during the exhaust process leads to greater pumping losses, as the piston must do more work in displacing the exhaust gases. This can potentially be offset by the work obtained from the turbocompounding turbine.

Figure 2. Turbocompounding Setup


Figure 3 shows the flowchart based on which the turbocompounding simulation is performed. In the turbocompounding case, three values (60%, 70%, 80%) were assumed as efficiencies of the turbocompounding turbine and turbocharger unit. Suitable values are assumed for the back pressure and temperature at turbocompounding turbine inlet (P2 and T2). Using T2, an initial value for back pressure at the turbocompounding turbine inlet (P2,NEW) is calculated using Equation (11). If the difference between P2, NEW and P2 is greater than or equal to 0.01, then the loop proceeds and assigns the value of P2, NEW to P2 (which is the variable of reference for the loop to continue). Using P2, NEW and the isentropic efficiency of the turbocompounding turbine, the value of temperature at the turbocompounding turbine inlet (T2, NEW) can be calculated. By using T2, NEW to calculate P2, NEW, if the difference between P2,NEW

and P2 is greater than 0.01, the loop continues; otherwise the work done by the turbocompounding turbine is calculated using Equation (12). This work done is added to the crankshaft work to recalculate the net IFCE of the turbocompounded engine by multiplying it with appropriate mechanical efficiency. Nomenclature for the flowchart: hTC – Turbocompound turbine efficiency hTT – Turbocharger efficiency TT – Turbocharger turbine TC – Turbocompounding turbine P2 – Assumed back pressure at TT inlet P2, NEW – Calculated back pressure at TT inlet T2, NEW- calculated temperature at TT inlet P3 – intake boost pressure WTC – work done by TC turbine T2- assumed temperature at TT inlet

In Equation (11), the subscripts c and t represent compressor and turbine of the turbocharger unit, respectively. P4 refers to the pressure at the exit of the turbocharger unit (typically atmospheric pressure), P5 refers to the pressure of the air entering the compressor, P3 refers to the intake boost pressure. Equation (12) is the isentropic efficiency equation for a turbine, which has been rearranged to determine T2, NEW. In Equation (13), 𝑊��d� refers to work done per cycle by the engine, 𝑊 ¡ refers to the work extracted from the turbocompounding turbine,𝜂��e is the mechanical efficiency of the transmission of power from the turbocompounding turbine to the crankshaft, 𝑚brefers to the fuel added per cycle, 𝑄defis the lower heating value of the fuel.

P2, NEW and T2, NEW can be found by using the following formulae as specified by Heywood [14].

𝑃8,l�j =1𝑃£⎩⎪⎨

⎪⎧

1 −

⎣⎢⎢⎢⎡Q𝑃�𝑃�

R«�V�«� − 1

hT T

⎦⎥⎥⎥⎤𝑇�

𝑇8,l�j𝐶,,�𝐶,,r

⎭⎪⎬

⎪⎫

«²«²V�

(11)

Figure 3. Flow chart for Turbocompounding Model

𝑇8,l�j = 𝑇� − h ¡𝑇� ³1 − �𝑃8,l�j𝑃�

�«²V�«²

´(12)

The net IFCE is calculated using the following equation:

𝜂l�r*l)*�µr�)¶¡· =𝑊��d� + 𝜂��e𝑊 ¡

𝑚b𝑄def(13)

SIMULATION TEST MATRIX Table 2 provides details on the operating conditions for

which the simulations were carried out.

Table 2. Simulation Test Matrix SOI (CAD) BMEP (bar) TC SIMULATION*

355 3.38 Yes 350 3.40 Yes 340 3.32 Yes

*TC – Turbocompounding

RESULTS AND DISCUSSION For every operating condition, the open-cycle simulation

was performed for multiple “simulated engine cycles” until convergence was achieved for pressure and temperature at GETDC. Experimental in-cylinder pressure and AHRR data were obtained for 1000 consecutive engine cycles and ensemble averaged [12]. Simulation results were validated by comparing with experimental in-cylinder pressure and AHRR histories and mass trapped at IVC. Table 3 provides the results of trapped mass at IVC. The simulation was first validated with experimental results where the exhaust manifold pressure was atmospheric.

P2 = P2,NEW

START

FIND P2,NEW FOR DESIRED P3USING T2

ASSUME hTC , hTT, T2,P2

(P2,NEW –P2)>=0.01?

FIND WTC

USE P2,NEW TO FIND T2,NEW USING hTC

USE T2,NEW TO FIND P2,NEW

Y

N


Following these initial validations, the model was simulated for turbocompounding cases. Results are also presented in the same order.

Figure 4. Pressure Comparison 355 CAD SOI

Figures 4 and 5 show the comparison of experimental

pressure and heat release rates with those of simulated values. The peak pressure during combustion and the peak AHRR values are predicted to within 1% accuracy compared to the corresponding experimental values.

Figure 6 shows the increase of in-cylinder pressure due to higher exhaust manifold pressure, which leads to greater pumping losses. A clear distinction is observed in the mass flow rate curves between the case when the exhaust manifold pressure is increased and the exhaust manifold pressure is maintained at atmospheric conditions. When the manifold pressure is set equal to in-cylinder pressure at EVO, as the piston moves down after EVO the in-cylinder pressure decreases (due to expansion) and falls below the exhaust manifold pressure; thus, some amount of air/exhaust gases enters the cylinder, as shown by the mass flow rate curve.

Table 3. Comparison between simulated and

experimental trapped mass – without Turbocompounding SOI (CAD)

EXPERIMENTAL TRAPPED MASS (g)

SIMULATED TRAPPED MASS (g)

355 2.632 2.687 350 2.638 2.608 340 2.654 2.577

This appears as a negative mass flow rate in Figure 6. The negative mass flow rate means that mass is entering the cylinder, as mass leaving the cylinder is considered to represent a positive rate. Around 540 CAD, exhaust gases start leaving the cylinder due to displacement by the piston, thus eliminating the blowdown process. Exhaust gases leaving the cylinder at this time pass through the turbocompounding turbine.

Figure 5. AHRR Comparison – 355° SOI

Figure 6. Exhaust Parameters – Turbocompounding

355° SOI Key – Figures 6 and 7 In-cyl Pr – In cylinder pressure in bar MFR – Mass flow rate into/out of the cylinder in kg/s Pex – Exhaust manifold pressure in bar Pevo – In cylinder pressure at EVO in bar

Figure 7 shows the log P vs log(V/Vmax) diagram, where P refers to the predicted in-cylinder pressure, V refers to cylinder volume at a particular crank angle, and Vmax refers to the maximum cylinder volume. The increase in the area of the pumping loop (circled) can be observed as the exhaust manifold pressure is increased. Figures 8 and 9 show the improvement in net IFCE when the turbocompounding turbine efficency as well as the overall turbocharger efficiency were varied from 60% to 80% for various injection timings. A mechanical efficiency of 90% was assumed for the transmission of power between the turbocompounding turbine and the engine crankshaft. A considerable increase was observed in net IFCE when the model

-2

0

2

4

6

8

10

12

-2

0

2

4

6

8

10

12

500 550 600 650 700 750

Valv

e Li

ft (m

m)

CAD

In-cyl Pr (Pex=Pevo) (bar)

MFR *10 (Pex=Pevo)(kg/s)

In-cyl Pr (Pex=1 bar) (bar)

MFR*10 (Pex = 1 bar) (kg/s)

Valve Lift


was simulated for turbocompounding conditions. For 355 CAD SOI, the net IFCE improved from a baseline value of 25% to 37%, when the turbocompounding turbine as well as the turbocharger operated at 80% efficiency. Similar improvements were found for other SOIs as well (e.g., for 340 CAD SOI as shown in Figure 9).

Figure 7. log P vs log (V/Vmax) for various exhaust

manifold pressures

CONCLUSIONS An open cycle thermodynamic simulation was performed

for a 4-stroke CI single cylinder engine, operating on diesel-ignited natural gas dual fuel combustion. The validated simulation was used to explore higher exhaust manifold pressures in an effort to eliminate the exhaust blowdown process and recover exhaust exergy using turbocompounding. The following conclusions can be drawn from the results presented in this paper:

Figure 8. Net Indicated Fuel Conversion Efficiency

Improvement – Turbocompounding (355° SOI)

Figure 9. Net Indicated Fuel Conversion Efficiency

Improvement – Turbocompounding (340° SOI)

1. Experimental pressure and trapped mass at IVC were predicted to within 1 % error, and the net-indicated fuel conversion efficiency was predicted to within 9-12 % error compared to the corresponding experimental values for an exhaust manifold pressure of 1 bar and an intake boost pressure of 1.48 bar.

2. Apparent heat release rates were determined by using a double – Wiebe (Miyamoto model) fit and yielded excellent AHRR predictions.

3. Increasing exhaust manifold pressure increases the pumping losses during the exhaust process but also results in greater potential for exhaust energy recovery via turbocompounding, which can more than compensate for the pumping losses.

4. Net indicated fuel conversion efficiencies improved from 25% to 37% after turbocompounding for 355CAD SOI when both the turbocompounding turbine and the turbocharger setup were operating at 80% efficiencies. Efficiencies improved significantly for other SOIs as well.

Finally, open cycle thermodynamic simulations, apart from providing an insight into the physics behind engine processes, can also help researchers in conducting several such hypothetical “thought” experiments and parametric studies aimed at improving engine efficiencies and emissions.

ACKNOWLEDGEMENTS Our sincere thanks to SASTRA University, Thanjavur, India

for providing us the opportunity to do our senior year project at The University of Alabama, Tuscaloosa, USA through their Semester Abroad Program (SAP). Adithya Baburaj and Abhinandhan Narayanan acknowledge the support provided by the Department of Mechanical Engineering at The University of Alabama during their internship in Spring 2018. The authors thank Mr. Hamidreza Mahabadipour, Mr. Prabhat Ranjan Jha,

ηTC =0.8

ηTC =0.7

ηTC =0.6


Mr. Kendyl Patridge for their assistance with the experimental data.

REFERENCES [1] Hamidreza Mahabadipour, Kalyan Kumar Srinivasan, Sundar Rajan Krishnan, Swami Nathan Subramanian (2018), Crank angle – resolved exergy analysis of exhaust flows in a diesel engine from the perspective of exhaust waste energy recovery, Applied Energy, 218, 31-44. [2] Aoyama T, Hattori Y, Mizuta JI, Sato Y. (1996) An experimental study on premixed-charge compression ignition gasoline engine, SAE Paper No. 960081. [3] Najit, P.M. and Foster., D.E., (1983), Compression-Ignited Homogeneous Charge Combustion, SAE Paper No. 830264. [4] Karim, G. A. (1968), Combustion in Dual Fuel Engines – A Status Report, Eighth International Conference on Combustion Engines, Congress Palace, Brussels, 6th-10th May 1968. [5] Yousuf Ismail et al (2015), A methodology for evaluating the turbocompound potential for an automotive engine, Proceedings of the Instituition of Mechanical Engineers,Part D: Journal of Automotive Engineering, 229, 1878-1893, DOI: https://doi.org/10.1177/0954407015572709. [6] Jerald A. Caton (2016), An Introduction to Thermodynamic Cycle Simulations for Internal Combustion Engines, John Wiley & Sons Ltd. [7] Patterson, D. J. and van Wylen, G. (1964), A digital computer simulation for spark-ignited engine cycles, in Digital Calculations of Engine Cycles, SAE Paper 633F. [8] McAulay et al (1965). Development and evaluation of the simulation of the compression-ignition engine, Society of Automotive Engineers, SAE paper no. 65045. [9] Krieger, R. B. and Borman, G. L. (1966). The computation of apparent heat release for internal combustion engines, American Society of Mechanical Engineers, ASME paper no. 66WA/DGP-4. [10] Lavoie, G.A et al (1970), Experimental and theoretical study of nitric oxide formation in internal combustion engines, Combustion Science and Technology, 1,313. [11] Heywood, J. B., Higgins, J. M., Watts, P. A., and Tabaczynski, R. J. (1979), Development and use of a cycle simulation to predict SI engine efficiency and NOx emissions, SAE Paper No. 790291. [12] Raihan et al. (2014), Experimental Analysis of Diesel Ignited Methane Dual-Fuel Low-Temperature Combustion in a Single Cylinder Diesel Engine, ASCE Journal of Energy Engineering, 141,2, p. C4014007, DOI: 10.1061/(ASCE)EY.1943-7897.0000235. [13] Zucrow, M. and Hoffman, J. D (1976), Gas dynamics, Vol.1, Page 55, Wiley, New York. [14] John B. Heywood (1988), Internal Combustion Engine Fundamentals,Appendix – C, Mc-Graw Hill, New York. [15] Krishnan (2001), Heat Release Analysis of Dual Fuel Combustion in a Direct Injection Compression Ignition Engine, M.S. Thesis, The University of Alabama,USA.

[16] Miyamoto, N., Chikahisa, T., Murayama, T., and Sawyer, R. (1985), Description and Analysis of Diesel Engine Rate of Combustion and Performance Using Wiebe’s Functions, SAE Paper 850107. [17] Woschni, G. (1967), Universally Applicable Equation for Instantaneous Heat Transfer Coefficient in Internal Combustion Engine, SAE paper 670931. [18] Cho, H., Krishnan, S.R., Luck,R. and Srinivasan K.K. (2009), Comprehensive uncertainty analysis of Wiebe function – based combustion model for pilot – ignited natural gas engines, Proceedings of the Institution of Mechanical Engineers Part D Journal of Automobile Engineering, 223, 1481-1498, DOI: 10.1243/09544070JAUTO1103.

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