Multi-Temperature, Thermal & Ion Fraction Effects over ...

1 University of Washington - Computational Fluid Dynamics Laboratory ICOPS2000

Multi-Temperature, Thermal & Ion Fraction Effects over Wedge and

Bluff Body Shapes in Hypervelocity Flow

Ward W. Vuillemot, Uri Shumlak


Abstract

We numerically investigate the effect of ionization on hypersonic flows by using an approximate Riemann fully three-dimensional MHD solver, WARP3, which will include multiple temperature in the near future. The code calculates the ionization fraction as a function of temperature. We treat the flow as a single fluid with three (3) constituents, or namely ions, electrons, and neutrals.

It is believed that the inclusion of multi-temperatures effects may explain the experimentally measured increase of shock stand-off distance encountered when an ionized hypersonic flow stagnates bluff body.


Motivation

Current MHD code assumes a fully, singlely ionized fluid, where ion temperature is equal to electron temperature. While this simplifies the system of equations, it does not provide an adequate means to describe flow regimes where full ionization cannot be assumed.

These said flow regimes include, but are not limited to: • hypersonic vehicles similar to National Aerospace Plane • ballistic re-entry vehicles • advanced plasma thrusters for plasma thrusters• portable, pulsed power systems


Generalized MHD Equations (0 <= fi <= 1)

• In order to modify WARP3 to include single fluid, multiple constituent flow, the single fluid equations have been re-derived to include ion fraction, fi, effects.

• We begin by assuming that ionization level, Z, is always of unity. Consequently, the number densities relations between ions, electrons, and neutrals are readily obtained. – nions = nelectrons = fionsn – nneutrals = (1 - fions)n


Non-dimensional, Conserved MHD Equations

• Continuity Equation

• Conservation of Momentum

• Magnetic Pressure

• Energy Conservation

�

∂ρ∂t

+ ∇⋅ ρ��

V = 0

�

∂ρ��

V ∂t

+ ∇⋅(ρ��

V ��

V − fi��

B ��

B + (p + fi��

B ⋅��

B 2

)��

I = ρ ˆ ν visc

�

∂��

B ∂t

+ ∇ ⋅ fi��

V ��

B −��

B ��

V ( )= ˆ B res + ˆ B Hall + ˆ B diamag

�

∂e∂t

+ ∇⋅ e + p +��

B ⋅��

B 2

��

V −��

B ��

V ( )��B

=

ˆ B res + ˆ B Hall + ˆ B diamag


Generalized Ohm’s Law

• Following the prototypical methodology utilized to derive the generalized Ohm's law from two fluid equations, it can shown that the ion fraction effects are:

��

��

E +��

U ×��

B = 1ne

H − Dfi

− Rmions fi

− 1− fi( )N

LHSE, electric field

U, velocity field

B, magnetic field

RHSH, Hall term

D, Diamagnetic term

R, Resistive term (electron—ion collision term)

N, Neutral Collision term


Generalized Ohm’s Law (details)

• Resistive

• Neutral Collision

• Hall1

• Diamagnetic1

1: Not Supported in WARP3 due to dependence on small time scales in comparison to other terms � computationally costly. Typically they are assumed to be too small to affect our solution

Cn2me ueme

mi

12

− ui

�

��

J ×��

B

�

∇⋅��

P e

, C is a collision frequencyconstant, [L3t-1]

�

∇⋅��η ⋅ ∇ ×

��

B ( )×��

B


Explanation of Various Effects

• Thermal Effects– We can expect these effects to diffuse out, forcing transport of

energy from energy-rich regions to energy-poor regions. We should not observe significant effects for hyperbolic systems (e.g. fluid speed greater than sonic speed.)

• Ion Fraction Effects– An ion fraction less than one will adversely effect the effective

magnetic field strength by the square root of the ion fraction.

– We can also expect the resistive diffusion to increase with a decrease in the ion fraction. We assume that the resistive diffusion will go to the vacuum resistive value asymptotically.


Explanation of Various Effects (continued)

• Multi-Temperature Effects– A model for a single fluid composed of three constituents

(neutrals, ions, and electrons), will naturally have multiple methods of energy transfer for different heating mechanisms.

– Heating Mechanisms & Associated Constituents• Viscosity

– Neutrals

– Ions

• Resistivity & Radiation– Electrons

• Thermal Conduction– Electrons

– Neutrals

– ions


Wedge Flow

Region 1 (Freestream) Conditions• Mach #: 3• Pressure: 0.714• Pressure: 1.0• γ: 1.4

Region 2 — Comparison of Analytical Results to Computational Results forSupersonic Flow over Wedge (δ = 25°)

δδδδ θθθθ Density X-velocity Y-velocity Pressure

Analytical 25 44.5 2.796 2.279 0 3.517

Computational 25 44.3 2.790 2.276 1.0d-4 3.517

Wedge

Flow 1 2

Shock

δθ


0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.1

0.2

0.3

Mach 3.0, 25 degrees

Wedge Flow : Computational Results

• Density contours

• Velocity profiles streamtraces

• Flow enters at 25º angle from left and top boundaries, equivalent to a 25ºwedge grid, but finite volume grid is simpler to generate


Bluff Body Flow

• Nitrogen (N2) gas bluff body flow simulated at 50,000 feet standard atmosphere (density = 0.186 kg m-3, temperature = 217 K) for the following Mach numbers: 0.25; 0.5; 1.0; 1.5; 3.0; and 3.0. At this altitude, the sonic velocity is approximately 300 m s-1.

• A case of increased enthalpy at Mach 3.0 for temperature of 5000 K. At this Mach number and increased temperature, the sonic velocity is approximately 460 m s-1.– Pressure is varied so that density matches standard atmosphere at

50,000 feet and temperature matches artificially high enthalpy

– The exact mechanism to induce this increased enthalpy is not of concern, however something similar to K. Chadwick et al tuned frequency waveguide might be applicable


Bluff Body Flow : Generalized Grid

• Use of generalized, algebraic grid for our initial study helps validate results. Bow shock formation is not a consequence of the grid, but of physics.

• Future studies will use elliptic grid to minimize computational costs associated with the freestream flow region


Bluff Body Flow : Drag

Mach[ # ]

SonicSpeed[ m s-1 ]

Density[ kg m-3 ]

Pressure[ Pa ]

ThermalEnergy[ °K ]

Drag1

0.25 75 0.186 11,600 217 2.10

0.5 150 0.186 11,600 217 1.94

1.0 300 0.186 11,600 217 1.00

1.5 450 0.186 11,600 217 1.22

3.0 900 0.186 11,600 217 2.01

3.0 1,441 0.186 276,000 5,000 48

1 : Normalized to sonic drag value (Mach # = 1)

Drag versus Mach #Standard Atmosphere Conditions

0

0.5

1

1.5

2

2.5

0.25 0.5 1 1.5 3Mach #

NormalizedDrag

• Pressure Drag along the hemisphere is lowest at sonic velocity, where the bow shock is strongest

Calculated for Standard Atmosphere at 50,000 feet unless otherwise noted


Bluff Body Flow : Blast Theory

• WARP3 results coincide with first-order approximation Blast Theory prediction

• Shock stand-off distance is chosen to match WARP3 results, as there is no theory that accurately predicts distance

Blast Theory Prediction at Mach 3.0

• Density contours– red-green interface represent

approximate stagnation region

• Yellow line represents Blast theory prediction


Bluff Body Flow : Subsonic

• Density Contours

• Velocity Profile Streamtrace

rho0.19

0.19

0.19

0.18

0.18

0.18

0.18

0.18

Mach 0.25

rho0.20

0.20

0.19

0.18

0.18

0.17

0.17

0.16

Mach 0.5


Bluff Body Flow : Sonic & Supersonic

• Density Contours


rho0.26

0.24

0.22

0.20

0.18

0.15

0.13

0.11

Mach 1.0

rho0.39

0.35

0.31

0.28

0.24

0.20

0.17

0.13

Mach 1.5


Bluff Body Flow : Mach 3 Comparison• Density

Contours


• Ion Fraction Contours


rho0.66

0.58

0.51

0.44

0.36

0.29

0.22

0.14

Mach 3.0, T = 5000 K & T = 217 K

fi0.04

0.03

0.03

0.02

0.02

0.01

0.01

0.00

Mach 3.0, T = 5000 K


Conclusions

• Present state of WARP3 has been benchmarked to analytic results for wedge flow and first-order approximation blast theory for bluff body flow

• WARP3 is a robust CFD tool that includes such effects as: resistive, viscous, and thermal diffusion, and ion fraction effects

• Relatively small increase in enthalpy results in significant gains in ion fraction at stagnation region

• In the future, we will combine codes so that WARP3 will be an explicit/implicit MHD solver with three-constituent single-fluid flow


Conclusions (continued)

• There is evidence from experiments both in the United States and former Soviet Union that an ionized medium can reduce the drag on hypersonic vehicles. – These effects may be explained by differences in the three

constituents' temperatures, and subsequent coupled, energy transport evolution.

• Multiple temperature and ion fraction effects coupled with electrically conducting fluid may results in many benefits in such diverse areas as: – reduced wave drag

– reduced skin friction

– reduced control-effector cross-section and/or divergement angle

– vectorable thrust


Poster Talk Request

• You can download a PDF version of this poster talk. It can be found at:

<http://www.aa.washington.edu/cfdlab/publications/vuillemot,_ward/icopsconf.pdf>

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