ContentServer (1)

NUMERICAL INVESTIGATION OF LAMINAR-PLATETRANSPIRATION COOLING BY THE PRECONDITIONEDDENSITY-BASED ALGORITHM

H. J. Zhang and Z. P. ZouNational Key Laboratory of Science and Technology on Aero-Engines,School of Jet Propulsion, Beihang University, Beijing, China

The preconditioned density-based algorithm and two-domain approach were used to inves-

tigate the laminar-plate transpiration cooling process. In the porous zone, the momentum

equations were formulated by the Darcy-Brinkman-Forchheimer model; and the local

thermal equilibrium model (LTE) was adopted for the energy equation. At the porous/fluid

interface, the stress-continuity interfacial condition was utilized. The effects of coolant

injection rate, Reynolds number, pressure gradient of mainstream, and thermal conductivity

ratio on the transpiration cooling performance were studied. Results indicate that the

thickness of the coolant boundary layer on the protected surface has a significant effect

on the transpiration cooling effect. Under the current conditions, the transpiration cooling

performance would be enhanced with increasing coolant injection rate, but would be

weakened with increasing Re. The adverse pressure gradient of mainstream would improve

the transpiration cooling performance.

1. INTRODUCTION

With the development of aerospace technology, the thermal loading imposedon the hot-end components of aerospace vehicles is becoming higher and higher.Thermo-protection technologies are faced with more and more austere challenges.Due to unique advantages such as high cooling efficiency, good covering abilityand easy control, the transpiration cooling based on a porous matrix is one of themost efficient cooling technologies to prevent surface damage from hot gas.

Many researchers adopt experimental, analytical or numerical methods toinvestigate transpiration cooling. Kays et al. [1–4] conducted systematic experimentalstudies to forced convective turbulent-plate transpiration cooling, and influence of theinjection rate, suction rate, and pressure gradient on cooling performance were inves-tigated. The experimental results agree well with the analytic dimensionless Stanton

Received 16 May 2012; accepted 30 June 2012.

This work was financially supported by the National Nature Science Foundation of China under

grant number 50776003; the Innovation Foundation of BUAA for Ph.D. graduates

(YWF-12-RBYJ-010); and the Specialized Research Fund for the Doctoral Program of Higher

Education (20101102110011).

Address correspondence to Z. P. Zou, National Key Laboratory of Science and Technology on

Aero-Engines, School of Jet Propulsion, Beihang University, Xueyuan Rd. No. 37, Beijing, Haidian

District, 100191, P.R. China. E-mail: [email protected]

Numerical Heat Transfer, Part A, 62: 761–779, 2012

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2012.712462

761

correlations proposed by Mickley et al. [5]. Liu and Chen [6], Thames and Landrum[7], and Frohlke et al. [8] studied the characteristics of transpiration cooling in theliquid rocket combustor and the exhaust nozzle, with the hydrogen utilized as the cool-ant medium. Choi et al. [9] investigated the feasibility of transpiration cooling in asupersonic ramjet combustor, with the helium as the coolingmedium instead of hydro-gen. Wang et al. [10, 11] studied the transpiration cooling experimentally in a hot windtunnel based on the background of gas turbine blade cooling.

Transpiration cooling includes two processes. First, the coolant medium flowsthrough the porous matrix and removes the heat flux by convection; then, the cool-ant medium interacts with the hot mainstream and forms a uniform coolant layer onthe hot-side surface. The interaction of the two processes plays an important role inthe transpiration cooling. The numerical investigations of transpiration cooling canbe divided into two categories: the coupling method and the un-coupling method. Asfor the un-coupling method, the fluid flow and heat transfer characteristics of acoolant medium in the porous matrix are studied by analytic method; and theinteractions of coolant flow and hot mainstream are studied by numerical or analyticmethods. Wang and Shi [12] investigated the transpiration cooling process using theun-coupling method. The flow in a porous region was simplified as one-dimensionalflow, and the local thermal nonequilibrium (LTNE) model was utilized for theenergy equation. Then, the error caused by the assumption of the local thermal equi-librium (LTE) model is discussed, and a quantitative criterion to choose the LTE or

NOMENCLATURE

Ar ratio of mainstream inlet area to outlet

area

Cs specific heat of solid, J=Kg K

CF inertia drag coefficient of porous media

Da Darcy number based on porous plate

length

E total energy per unit mass of fluid, J=Kg

F coolant flow injection rate~FF flux vector

H porous plate height, m

Hf total enthalpy per unit mass of fluid,

J=Kg

h porous layer thickness, m

hf static enthalpy per unit mass of fluid,

J=Kg

K permeability of porous medium, m2

L porous plate length, m

P fluid pressure, Pa

q heat flux density, W=mK~QQ source term

Re Reynolds number based on mainstream

inlet velocity

Rev Reynolds number based on coolant inlet

velocity

T temperature, K

s time, s

u, v, w velocity component, m=s~VV velocity vector, m=s

~WW vector of conservative variables

~WpWp vector of primitive variables

x, y, z coordinates

dij Kronecker delta function

e porosity of porous medium

h dimensionless temperature

k thermal conductivity, W=mK

m dynamic viscosity, Kg= ms

q density, Kg=m3

s viscous stress tensor

hi volume averaging

Superscripts and Subscripts

c related to convection

e effective

f fluid phase in porous medium

fl pure fluid

i the porous=fluid interface

in inlet

s solid phase in porous medium

v related to diffusion

w wall

762 H. J. ZHANG AND Z. P. ZOU

LTNE model is suggested. Polezhaev and Seliverstov [13] proposed a universalone-dimensional model for the fluid flow in the porous matrix, with the LTNE modeladopted for the energy equation. The specified heat flux density was given for theboundary condition of the hot-side surface. Wolfersdorf [14] and Wang and Shi[15] investigated the influence of the coolant side and hot side boundary conditionson transpiration cooling performance, respectively, with the LTNE model adoptedfor the energy equation. Elbashbeshy [16] and Aydin and Kaya [17] investigatedthe laminar boundary layer characteristics with uniform gas injection or suctionby the analytical method. Jiang et al. [18] studied the turbulent flow and heat transferin a rectangular channel with and without transpiration cooling by experimental andnumerical methods. The influences of injection rate on the convective heat transferrate and turbulent boundary layer characteristics were analyzed.

As for the coupling methods, Hwang and Chang [19] simulated the forced con-vective heat transfer in a square duct with one-wall injection and suction. In theporous region, the flow was considered to be one-dimensional, and the LTNE modelwas adopted for the energy equation. At the porous=fluid interface, the temperatureof the solid phase was assumed to be equal to that of the fluid phase. Yu and Jiang[20] applied the commercial CFD software Fluent to investigate the high tempera-ture transpiration cooling in a cylindrical porous channel, with the LTE modeladopted for the porous energy equation. The RNG k-e turbulence model was usedfor the simulation of turbulent mainstream flow. Liu et al. [21] investigated thetranspiration cooling mechanisms used for the thermal protection of a nose conefor various cooling gases by the coupling method based on Fluent. The numericalresults agree well with the experimental data.

There are many factors that affect the fluid flow and heat transfer characteristicsof transpiration cooling, and the parametric study was conducted here. In reference[22], a preconditioned density-based algorithm was proposed to solve the conjugatefluid flow and heat transfer problem in hybrid porous=fluid=solid domains. Thenumerical results indicate that the proposed numerical method has characteristicsof high numerical precision and good applicability. On the basis of this, thelaminar-plate transpiration cooling was investigated numerically. The effects of cool-ant injection rate, Reynolds number, pressure gradient of mainstream, and thermalconductivity ratio on the transpiration cooling performance were analyzed.

2. MATHEMATICAL MODEL

2.1. Problem Description

The schematic diagram of the computational model is shown in Figure 1. Thehot mainstream with uniform temperature Tin and velocity Uin flows across theporous plate with length L and thickness H. The coolant medium with low tempera-ture Tc is injected into the porous plate at constant velocity Vc. The coolant flowsthrough the porous matrix, and forms a uniform coolant layer over the hot-side sur-face of the porous structure. On one hand, the coolant flow removes the heat flux inthe porous matrix by convection; on the other hand, the coolant layer could isolatethe hot-side surface from hot mainstream, and increase the convective thermal resist-ance. So, the cooling efficient of transpiration cooling is very high.

LAMINAR-PLATE TRANSPIRATION COOLING 763

2.2. Governing Equations

In this article, the two-domain approach is adopted to simulate the fluid flowand heat transfer problems in porous=fluid hybrid domains, and each domain hasthe corresponding governing equations. It should be noted that the coolant mediumand mainstream are the same kind of fluid, and there is no phase transition.

For simplicity, only the governing equations for porous domain are given. Thefollowing assumptions are adopted in the models: 1) the porous medium is homo-geneous and isotropic; 2) the momentum equations were formulated by the Darcy-Brinkman-Forchheimer model; 3) the local thermal equilibrium model (LTE) wasadopted for the energy equation; 4) ignore the impact of temperature on the physicalproperties; and 5) the mainstream flow is laminar. Then, the macro-governing equa-tions for porous domain based on volumetric method could be written as follows.

qðehqif Þqs

þqðehqif hvifj Þ

qxj¼ 0 ð1Þ

qðehqif hvifi Þqs

þqðehqif hvifi hvi

fj Þ

qxj¼ � qðehPif Þ

qxiþ mf

qqxj

qðehvifi Þqxj

" #

� e2mfK

hvifi � e3 K�0:5CFqf h~VVif�� hvifi ð2Þ

qðehqif hEif þ ð1� eÞhqiscshTif Þqs

þqðehqif hHif hvifj Þ

qxj¼

qðhsifijhvifi Þ

qxj�qhqifjqxj

ð3Þ

Where e is the porosity; K is the permeability; CF is the dimensionless drag

coefficient; hEif is the average total energy per unit mass hEif ¼ hHif � hPif =hqif ;hHif ¼ hhif � jh~VVif j2=2, hhif ¼ hðhPif ; hTif Þ; mf q

qxj

qðehvifi Þqxj

h iis the Brinkman term;

and hqifj is the average heat conduction term hqifj ¼ �keqhTifqxj

, where the effective

thermal conductivity of porous medium ke could be simply estimated as:ke¼ ekfþ (1� e)ks, when the tortuosity and thermal dispersion are neglected. Thecorresponding fluid state equation: hqif¼ q(hPif, hTif). hqis, cs are density and spe-cific heat of solid, respectively. The physical parameters in the porous domain men-tioned above are all intrinsic-phase average values.

Figure 1. Schematic diagram of the computational model.


When the two-domain approach is applied to simulate the conjugate fluid flowand heat transfer problems in porous=fluid hybrid domains, the relevant boundaryconditions should be specified at the porous=fluid interfaces. The mass, momentum,and energy balance conditions should be satisfied at the interface. In this article, theshear stress continuity model is adopted for the interfacial hydrodynamic boundarycondition. In transpiration cooling, the flow in porous medium is low-speed seepageflow, and the development of thermal equilibrium between coolant medium andporous matrix is adequate enough. When the temperature difference between solidphase and fluid phase is not big, the local thermal equilibrium (LTE) model couldbe adopted for the energy equation in the porous domain. Correspondingly, thetemperature and heat flux continuity model are applied for the interfacial thermalboundary condition.

2.3. Numerical Method

In this article, the density-based finite-volume algorithm is used to simulate thefluid flow and heat transfer problem in hybrid porous=fluid domains, with the pre-conditioning method applied to solve the convergence and stability problems inthe calculation of the low-speed flows. Because the density-based algorithm in thepure fluid domain is commonly seen, only the numerical procedure for the porousdomain is presented. For the energy equation of the porous domain, the pseudo-timederivative term is very complicated and the derivation of the preconditioning matrixis difficult. For the time-marching method, the appropriate change to thepseudo-time derivative terms would not influence the final steady converged sol-ution. In this article, the pseudo-time derivative term of the energy equation (Eq.

(3)) qðehqif hEifþð1�eÞhqiscshTif Þqs is modified as qðehqif hEif Þ

qs . Then, the governing equations

of the porous domain (Eqs. (1)–(3)) can be expressed as follows.

qqs

ZX

~WW � dXþIqX

~FF c � ~FFv

� �� dS ¼

ZX

~QQ � dX ð4Þ

Where ~WW ¼ ½ehqif ehqif huif ehqif hvif ehqif hwif ehqif hEif �T is the vector

of conservative variables: ~FF c ¼ ½hqif h~VVi hqif huif h~VVi þ hPi~ii hqif hvif h~VVi þhPi~jj hqif hwif h~VVi þ hPi hqif hHif h~VVi�T is the vector of convective fluxes:~FFv ¼ ½0 hsxiif hsyiif hsziif hsijif hvjif þ hqiif �T is the vector of viscous fluxes: and ~QQ

is the source term.

The preconditioned N-S equations, formulated in primitive variables ~WWp, can

be written as follows.

C � qqt

ZX

~WWp � dXþIqX

~FF c � ~FFv

� �� dS ¼

ZX

~SS � dX ð5Þ

Where ~WWp is the vector of primitive variables. When choosing primitive vari-

ables in the porous domain, it should be under consideration that the flow physicalparameters should be continuous in space. According to this request and the primi-tive variables used in SIMPLE-family schemes in the porous domain, the vector of

primitive variables is chosen as: ~WWp ¼ ½hPif ehuif ehvif ehwif hTif �T ; ~FF c; ~FFv, and ~QQ


are consistent with the terms mentioned above; and C is the preconditioning matrix.In the clear fluid region, the preconditioning matrix proposed by Weiss and Smith[23] is applied. The preconditioning matrix for the porous domain can be seen in ref-erence [22]. Although it’s not the same as that for the clear fluid domain, the form ofeigenvalues of preconditioned convective flux Jacobian is the same as that for theclear fluid domain, which means that the proposed preconditioning matrix for theporous domain could alleviate the stiffness of governing equations in the porousdomain effectively, and also solve the convergence problem at low speed.

For the clear fluid or porous domains, the discretization schemes of the govern-ing equations are almost the same. The central scheme is used for the spatial discre-tization, with the matrix artificial dissipation scheme [24, 25] which has a smallernumerical dissipation applied. The detailed discretization scheme and numerical pro-cedure can be seen in reference [22].

2.4. Code Validation

The computational domain of transpiration cooling includes the porous andpure fluid domains. In references [22, 26], the numerical results indicate that thenumerical method adopted here is suitable for the simulation of the coupled fluid flowand heat transfer problems in hybrid porous=fluid=solid domains. In order to indicatethe reliability to the simulation of transpiration boundary layer flows, the present codewas validated by the case of laminar flow in a plate channel with permeable walls.

The schematic diagram of laminar flow in a plate channel with permeable wallsis presented in Figure 2, where the upper and bottom walls are permeable. The cool-ant flow enters into the mainstream from the bottom wall and outflow from theupper wall with the same velocity Vc. The flow is laminar and incompressible. Whenthe flow is under a fully developed condition, the velocity distribution is thesuperposition of a motion perpendicular to the walls with velocity Vc and a axialmotion with the velocity U(y) [27].

UðyÞ ¼ Dp �HVc

Y þ 1

shðRevÞ½chðRevÞ � expðY �RevÞ�

� �ð6Þ

Figure 2. Schematic diagram of laminar flow in a plate channel with permeable walls.


Where Ren ¼ VcHn ; Y¼ y=H; Dp ¼ � 1

qdpdx

Average channel velocity: Um ¼ 12H

RH

�H UðyÞdy ¼ Dp�HVc

chðRenÞshðRenÞ �

1Ren

h iCoolant injection rate: F¼Vc=Um

When Ren! 0, the flow mentioned above tends to be Poiseuille flow.The axial velocity profiles for different coolant injection rates are shown in

Figure 3, in comparison to the corresponding analytic solutions. It is found thatthe numerical results agree well with the analytic solutions for different injectionrates, which demonstrates the reliability of the present approach to deal with thetranspiration boundary layer flows. In addition, the axial velocity distributions tendto be more and more asymmetric with increasing coolant injection rates. Also, themaximum value of axial velocity increases and the location of maximum velocitymoves to the upper wall.

3. RESULTS AND DISCUSSION

The schematic diagrams of the computational grid are shown in Figure 4. Aftera mesh independent test, the grid density of the pure fluid domain is chosen as181� 41� 5, and the grid density of the porous domain is chosen as 101� 31� 5.The mesh is refined at the porous=fluid interface. The mainstream flow is laminar

Figure 3. Axial velocity distributions for different coolant injection rates (color figure available online).

Figure 4. Schematic diagrams of computational grid a) Overall grid and b) grid at the porous=fluid

interface (color figure available online).


and incompressible. The flow velocity and temperature are given for the mainstreamand coolant flow inlets. The static pressure is specified at the mainstream outlet. Theupper wall of the fluid domain is slip wall, and the bottom wall of the fluid domainwithout contact with the porous domain is the non-slip wall.

Besides geometry size, the dimensionless parameters controlling the transpi-ration cooling performance can be achieved with the governing equations nondimen-sionalized, which are: Reynolds number Re¼ qUinL=m; Prandtl number Pr; coolantflow injection rate F¼Vc=Uin; porosity e; Darcy number Da¼K=L2; and the ther-mal conductivity ratio ks=kf. In the following simulations, the above dimensionlessparameters are considered to be defaults without special emphasis, which are:Re¼ 5e4, Pr¼ 0.7, F¼ 0.1%, e¼ 0.5, Da¼ 5e-7, ks=kf¼ 10, and CF¼ 0.0. In this arti-cle, the influences of some parameters including Re, F, ks=kf, and the pressure gradi-ent of the mainstream on the transpiration cooling performance and transpirationboundary layer characteristics are studied.

3.1. Effect of Coolant Injection Rate F

The temperature contours for different coolant injection rates are shown inFigure 5. (The inlet temperature of the mainstream is 310 k; the inlet temperatureof coolant flow is 300 k.) With an increase in the coolant flow injection rate F, thetemperature in the porous domain and at the porous=fluid interface decreases.The coolant flow enters into the mainstream and forms a uniform coolant layer overthe hot-side surface of the porous structure. The transpiration boundary layerbecomes thicker, and the temperature and convective heat transfer coefficient atthe interface would decrease. At the downstream of the interface, the coolantboundary layer would develop and protect the downstream wall. But the cooling

Figure 5. Temperature contours for different coolant injection rates (color figure available online).


performance becomes worse with the heating of the mainstream (the downstreamnon-slip wall is adiabatic).

Figure 6 shows the dimensionless temperature h profiles at the porous=fluidinterface for different coolant injection rates. The dimensionless temperature isdefined as: h¼ (Ti�Tc)=(Tin�Tc), where Ti is the interfacial temperature; Tin isthe inlet temperature of mainstream; and Tc is the inlet temperature of coolant flow.It is obvious that the interfacial dimensional temperature decreases with an increasein coolant injection rate. Under the current conditions, the interfacial dimensionlesstemperature would decrease to 0.5 with the coolant flow injection F¼ 0.2%, whichdemonstrates the high efficiency of the transpiration cooling.

Figure 6. Dimensionless temperature profiles at porous=fluid interface for different coolant injection rates

(color figure available online).

Figure 7. Schematic diagrams of coolant flow streamline for different coolant injection rates (color figure

available online).


The schematic diagrams of coolant flow streamline for different injection ratesare presented in Figure 7. It can be seen that the coolant flow enters into the main-stream and forms a coolant boundary layer over the porous=fluid interface. At thedownstream, the boundary layer becomes thicker with the accumulation of the cool-ant flow. Besides, with an increase in the injection rate, the transpiration boundarylayer is thicker, and the transpiration cooling performance becomes better. Figure 8shows the velocity boundary layer thickness and thermal boundary layer thicknessprofiles at the porous=fluid interface for different coolant injection rates. It isobvious that the velocity and thermal boundary layers both become thicker with ahigher injection rate. The interfacial temperature and convective heat transfercoefficient would also decrease remarkably.

3.2. Effect of Reynolds Number Re

Obviously, the Reynolds number has a significant influence on transpirationboundary layer characteristics, so it would affect the transpiration cooling

Figure 8. a) Velocity boundary thickness and b) thermal boundary thickness profiles for different coolant

injection rates (color figure available online).


Figure 9. Dimensionless temperature profiles at the porous=fluid interface for different Re (color figure

available online).

Figure 10. a) Velocity boundary thickness and b) thermal boundary thickness profiles for different Re

(color figure available online).


performance. In this article, the change of Re is caused by the variation of themainstream inlet velocity. In order to maintain a constant injection rate F, the inletvelocity of coolant flow would change correspondingly.

Figure 9 shows the dimensionless temperature profiles at the porous=fluidinterface for different Re. The mainstream Re would affect the transpirationcooling performance remarkably. Under the current conditions, with an increasein Re the dimensionless temperature at the porous=fluid interface decreases, andthe transpiration cooling performance would be weakened. For the laminartranspiration boundary layer, the variation of Re would influence the boundarylayer characteristics. The effect of viscous force in the boundary layer wouldbe weakened and the boundary layer would become thinner with the increaseof Re; so, the transpiration cooling performance would become worse. In

Figure 11. a) Dimensionless axial velocity and b) dimensionless vertical velocity profiles at the porous=

fluid interface for different Re (color figure available online).


addition, when the Re is small, the transpiration cooling performance would bemore sensitive to the variation of Re.

The velocity boundary layer thickness and thermal boundary layer thicknessprofiles at the porous=fluid interface for different Re are shown in Figure 10. Withthe increase of Re, the thicknesses of the velocity boundary and thermal boundarylayers would decrease.

Figure 11 presents the dimensionless axial velocity and vertical velocityprofiles at the porous=fluid interface, where the axial velocity Vx is nondimensio-nalized by mainstream inlet velocity Uin; and the vertical velocity Vy is nondimen-sionalized by the coolant inlet velocity Vc. It is found that the variation of Re hasan obvious effect on the coolant velocity profiles at the porous=fluid interface. Atthe interface, the axial velocity increases to the maximum value from 0 at a shortdistance; then, it would decrease gradually. Besides, the interfacial coolant axialvelocity increases with an increase in Re. This is because the boundary layer

Figure 12. a) Interfacial dimensionless temperature and b) thermal boundary layer thickness profiles for

different ks=kf (color figure available online).


would become thinner, and the shear stress would increase with increasing Re. Itis known Figure 11b that the profile of dimensionless vertical velocity is not uni-form; and the nonuniform extent of interfacial vertical velocity would increasewith an increase in Re.

3.3. Effect of Thermal Conductivity ratio ks/kf

In this article, the fluid thermal conductivity is constant and the variation ofks=kf is caused by the change of solid thermal conductivity. Normally, the solid ther-mal conductivity is bigger than the fluid thermal conductivity, and the variation ofks=kf would influence the effect thermal conductivity of porous media (the LTEmodel is adopted for the energy equation). So, the thermal conductivity ratio hasa significant effect on the transpiration cooling performance. The interfacial dimen-sionless temperature and thermal boundary layer thickness profiles for differentks=kf are presented in Figure 12. It is found that the interfacial temperature decreasesand the thermal boundary layer becomes thicker with an increase in ks=kf. So, thetranspiration cooling performance would become better with increasing ks=kf.

3.4. Effect of Mainstream Pressure Gradient

The fluid flow characteristic in the porous domain is closely related to thepressure drop characteristic. So, the pressure distribution of mainstream wouldaffect the flow characteristic of porous domain. In this article, the influence ofdifferent mainstream pressure distributions on transpiration cooling performance

Figure 13. Pressure contours and coolant streamline distributions for different mainstream pressure dis-

tributions (color figure available online).


is analyzed. Three kinds of mainstream pressure distributions, including zero press-ure gradient, adverse pressure gradient, and favorable pressure gradient, are con-sidered. The variation of pressure distribution is caused by the change of themainstream channel shape. For the zero pressure gradient case, the mainstreamchannel would not change along flow direction. For adverse or favorable pressuregradient cases, the mainstream channel is divergent or convergent, respectively.The upper wall of the mainstream channel is a slip wall. Here, the constant velocityand temperature are given for the mainstream inlet condition, and the specified staticpressure for mainstream outlet is 1000 Pa. The boundary condition for the porousdomain would not change. In order to denote the strength of the pressure gradient,the ratio of the mainstream inlet area to outlet area (Ar) is used to demonstrate themainstream pressure distribution characteristics. Ar¼ 1 means zero pressuregradient; Ar< 1 means adverse pressure gradient; and the smaller Ar means biggeradverse pressure gradient. While, Ar> 1 is just the opposite.

Figure 14. a) Dimensionless temperature and b) friction coefficient profiles at the porous=fluid interface

for different Ar (color figure available online).


The pressure contours and coolant streamline distributions are presented inFigure 13 for the zero pressure gradient case, the adverse pressure gradient case,and the favorable pressure gradient case. It is found that the pressure distributionhas a significant influence on the pressure distribution in the porous domain andthe coolant streamline distribution. For the zero pressure gradient case(Figure 13a), the pressure distribution at the porous=fluid interface is relatively moreuniform, and the pressure gradient direction in the porous domain is almost vertical.So, the coolant streamline in the porous domain is almost vertical too, and the localcoolant injection rate on the hot-side surface is uniform. For the adverse pressuregradient case (Figure 13b), the pressure distribution in the porous domain changesobviously. The pressure gradient direction inclines toward the upstream, and so doesthe coolant streamline. The local interfacial coolant injection rate is not uniform, andthe local injection rate at upstream is relatively higher. Besides, the coolant boundarylayer would become thicker under adverse pressure gradient conditions, so the

Figure 15. a) Velocity boundary layer thickness and b) thermal boundary layer thickness profiles for

different Ar (color figure available online).


transpiration cooling performance would be better. For the favorable pressure gradi-ent case (Figure 13c), the phenomenon is just the opposite.

The dimensionless temperature and friction coefficient profiles at the porous=fluid interface for different Ar are shown in Figure 14. In Figure 14a it can be seenthat the mainstream pressure distribution has obvious effects on the interfacialtemperature and friction coefficient profiles. The interfacial temperature increaseswith an increase in Ar, which demonstrates that the mainstream adverse pressuregradient is beneficial to improve the transpiration cooling performance. Figure 13shows that the local coolant injection rate is higher at upstream under the adversepressure gradient condition, which could enhance the transpiration cooling perfor-mance. Besides, the mainstream adverse pressure gradient could make the coolantboundary layer become thicker, which would increase the convective thermal resist-ance between the hot mainstream and the protected surface. So, the existence ofthe mainstream pressure gradient would enhance the transpiration cooling perfor-mance. Figure 14b shows that the friction coefficient increases with an increase inAr. This is because the boundary layer would become thinner under the favorablepressure gradient, and the coefficient friction would increase. In addition, for the caseof Ar¼ 0.75, the boundary layer separation occurs at the location x� 0.85. Figure 15shows the velocity boundary layer thickness and thermal boundary layer thicknessprofiles at the porous=fluid interface for different Ar. Both the velocity boundarylayer thickness and thermal boundary layer thickness decrease with an increase in Ar.

4. CONCLUSION

The investigation of laminar plate transpiration cooling was conducted withthe two-domain approach and the preconditioned density-based algorithm adopted.The influences of some parameters such as coolant injection rate, Reynolds number,thermal conductivity ratio, and mainstream pressure gradient on the transpirationcooling performance were analyzed. The following conclusions can be drawn.

1. The coolant injection rate has a significant influence on the transpiration bound-ary layer characteristics. With the increase of injection rate, the coolant boundarylayer over a hot-side surface would become thicker, which would help to isolatethe protected surface to the hot mainstream and increase the convective thermalresistance. In the current conditions, the interfacial dimensionless temperaturewould decrease to 0.5 with a coolant injection rate of less than 0.2%.

2. The Reynolds number could influence the development of the coolant boundarylayer, and has obvious effects on the transpiration cooling performance. With theincrease of Re, the influence of viscous force would be weakened, and the coolantboundary layer would become thinner. So, the interfacial convective heat transfercoefficient would increase, and the transpiration cooling performance wouldbecome worse.

3. The mainstream pressure distribution could influence the coolant flow in theporous domain and the transpiration boundary layer characteristics, so it wouldaffect the transpiration cooling performance. With the existence of the main-stream adverse pressure gradient, on one hand, the coolant streamline wouldincline toward upstream, and the local coolant injection rate at upstream is


higher; on the other hand, the coolant layer over the protected surface wouldbecome thicker, and the convective thermal resistance would increase. The twoeffects would enhance the transpiration cooling performance. On the contrary,the mainstream favorable pressure gradient would weaken the transpiration cool-ing performance.

REFERENCES

1. R. J. Moffat and W. M. Kays, The Turbulent Boundary Layer on a Porous Plate:Experimental Heat Transfer with Uniform Blowing and Suction, Int. J. Heat MassTransfer, vol. 11, pp. 1547–1566, 1968.

2. W. M. Kays, Heat Transfer to the Transpired Turbulent Boundary Layer, Int. J. HeatMass Transfer, vol. 15, pp. 1023–1044, 1971.

3. D. W. Kearney, W. M. Kays, and R. J. Moffat, Heat Transfer to a Strong AcceleratedTurbulent Boundary Layer: Some Experimental Results, Int. J. Heat Mass Transfer,vol. 16, pp. 1289–1305, 1972.

4. P. S. Andersen, W. M. Kays, and R. J. Moffat, Experimental Results for the TranspiredTurbulent Layer in an Adverse Pressure Gradient, J. Fluid Mech., vol. 16, pp. 1289–1305,1972.

5. H. S. Mickley, R. C. Ross, A. L. Squyers, and W. E. Stewart, Heat, Mass and MomentumTransfer for Flow over a Flat Plate with Blowing or Suction, Rep. no. NASA -TN-3208,1954.

6. W. Q. Liu and Q. Z. Chen, Transpiration Cooling of Rocket Thrust Chamber with LiquidOxygen, AIAA-1998-890, 1998.

7. M. Thames and D. B. Landrum, Thermal=Fluid Study of Perforated Plates forTranspiration Cooled Rocket Chambers, AIAA-1998-3442, 1998.

8. K. Frohlke, O. J. Haidn, and E. Serbest, New Experimental Results on TranspirationCooling for Hydrogen=Oxygen Rocket Combustion Chambers, AIAA-1998-3443, 1998.

9. S. H. Choi, S. J. Seotti, K. D. Song, and H. Reis, Transpiration Cooling of a Scram JetEngine Combustion Chamber, AIAA-1997-2576, 1997.

10. J. H. Wang, J. Messner, and H. Stetter, An Experimental Investigation of TranspirationCooling, Part I—Application of an Infrared Measurement Technique, Int. J. Rot. Mach.,vol. 9, no. 3, pp. 1542–3034, 2003.

11. J. H. Wang, J. Messner, and H. Stetter, An Experimental Investigation of TranspirationCooling, Part II—Comparison of Cooling Method and Media, Int. J. Rot. Mach., vol. 10,no. 5, pp. 1542–3034, 2004.

12. J. H. Wang and J. X. Shi, A Discussion of Transpiration Cooling Problem through anAnalytical Solution of Local Thermal Nonequilibrium Model, ASME J. Heat Transfer,vol. 128, pp. 1093–1098, 2008.

13. Yu. V. Polezhaev and E. M. Seliverstov, A Universal Model of Heat Transfer in Systemswith Penetration Cooling, High Temp., vol. 40, no. 6, pp. 856–864, 2002.

14. J. Von Wolfersdorf, Effect of Coolant Side Heat Transfer on Transpiration Cooling, HeatMass Transfer, vol. 41, no. 4, pp. 327–337, 2005.

15. J. H. Wang and J. X. Shi, A Discussion of Boundary Conditions of Transpiration CoolingProblems using Analytical Solution of LTNE Model, ASME J. Heat Transfer, vol. 130,pp. 1–5, 2008.

16. E. M. A. Elbashbeshy, Laminar Mixed Convection over Horizontal Flat Plate Embeddedin a Non-Darcian Porous Medium with Suction and Injection, Appl. Math. Comp., vol.121, pp. 123–128, 2001.


17. O. Aydin and A. Kaya, Laminar Boundary Layer Flow over a Horizontal Permeable FlatPlate, Appl. Math. Comp., vol. 161, pp. 229–240, 2005.

18. P. X. Jiang, L. Yu, J. G. Sun, and J. Wang, Experimental and Numerical Investigation ofConvection Heat Transfer in Transpiration Cooling, Appl. Therm. Eng., vol. 24, pp. 1271–1289, 2004.

19. G. J. Hwang and Y. C. Chang, Developing Laminar Flow and Heat Transfer in a SquareDuct with One-Walled Injection and Suction, Int. J. Heat Mass Transfer, vol. 36, no. 9,pp. 2429–2440, 1993.

20. M. Yu and P. X. Jiang, Numerical Simulation of High Temperature Transpiration Cool-ing in Cylindrical Porous Channels, J. Tsinghua Univ. (Sci. & Tech.), vol. 46, no. 2, pp.242–246, 2006 (in Chinese).

21. Y. Q. Liu, P. X. Jiang, S. S. Jin, and J. G. Sun, Transpiration Cooling of a Nose Cone byVarious Foreign Gases, Int. J. Heat Mass Transfer, vol. 53, pp. 5364–5372, 2010.

22. H. J. Zhang, Z. P. Zou, Y. Li, and J. Ye, Preconditioned Density-Based Algorithm forConjugate Porous=Fluid=Solid Domains, Numer. heat transfer A, vol. 60, pp. 129–153,2011.

23. J. M. Weiss and W. A. Smith, Preconditioning Applied to Variable and Constant DensityFlows, AIAA J., vol. 33, pp. 2050–2057, 1995.

24. R. C. Swanson and E. Turkel, On Central Difference and Upwind Schemes, J. Comput.Phys., vol. 101, pp. 292–306, 1992.

25. R. C. Swanson, R. Radespiel, and E. Turkel, Comparison of Central Several DissipationAlgorithms for Central Difference Schemes, AIAA-97-1945, 1997.

26. H. J. Zhang and Z. P. Zou, Investigation of a Confined Laminar Impinging Jet on a Platewith a Porous Layer using the Preconditioned Density-Based Algorithm, Numer. HeatTransfer A, vol. 61, pp. 241–267, 2012.

27. N. V. Nikitin and A. A. Pavel’ev, Turbulent Flow in a Channel with Permeable Walls,Direct Numerical Simulation and Results of Three-Parameter Model, Fluid Dyn., vol.33, pp. 826–832, 1998.


Copyright of Numerical Heat Transfer: Part A -- Applications is the property of Taylor & Francis Ltd and its

content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's

express written permission. However, users may print, download, or email articles for individual use.

ContentServer (1)

Documents

Transcript of ContentServer (1)