Computational Fluid Dynamic Solutions of Hyper Sonic Viscous Flows

8/8/2019 Computational Fluid Dynamic Solutions of Hyper Sonic Viscous Flows

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Computational Fluid Dynamic Solutions of

Hypersonic Viscous FlowsHusein Bhinderwala

Student, Mechanical Dept., K.J.Somaiya College of Engineering

Mumbai University [email protected]

Abstract ² Our objective here will be to present various

approaches to the solutions of hypersonic viscous flows

which go beyond, and are more ³exact´ than, the

boundary layer analysis.

I. INTRODUCTION

Computational fluid dynamics (CFD) is the numerical

simulation of flowfields through the approximate solution of

the governing partial differential equations for mass,

momentum, and energy conservation coupled with the

appropriate relations for thermodynamic and transport properties. Today using CFD, the viscous interaction effect

can be calculated by treating the entire flowfield between the

body and shock as fully viscous² no division between a

boundary layer and a inviscid flow is made.

Another reason to apply a wholly viscous shock-layer

analysis over the conventional approach is that the derivation

of boundary layer equations does not provide for a way to

compute -- y-momentum.

The most accurate approach to the solution of a fully

viscous flow is the complete Navier-Stokes equations with no

basic simplifications whatsoever. It also allows the detailed

calculations of the complete flowfield over a body where the

flow is assumed to be viscous at every point. Hence it provides

everything about the flow, such as the shock shape, detailed

flow variables between the shock and the body, skin friction,

heat transfer, lift, drag, moments, etc. Finally we are going to

provide a CFD analysis done by the author, on a c-d nozzle for

solutions of hypersonic flow using the Zeus Numerix software

II.FULL NAVIER-STOKES SOLUTIONS

The ultimate in hypersonic viscous flow calculations is the

solution of the complete Navier-Stokes equations(Ref 1) withno reduction of any terms. The modern techniques of CFD in

combination with new supercomputers now allow the

numerical solution of the equations.

On close examination of Navier-Stokes equations, it is

concluded that they are a system of p.d.e.s with somewhat

mixed hyperbolic, parabolic and elliptic behaviour. Let uswrite the equations such that the time derivatives are on the

left side and all spatial derivatives are on the right side of the

equations

Continuity equation

(1)

X Momentum equation

(2)

Y Momentum equation

(3)

Z Momentum equation

(4)

Energy equation

(5)

The time-marching solution of these equation isconceptually carried out as follows:

1. Cover the flowfield with grid point, and assume arbitrary

values of all the dependent variables at each grid point. This

represents the assumed initial condition at time t=0.

2. Calculate the values of p, u, v, w and (e+) from eqs.(1-5) as function of time, using a time marching finitedifference method. One such method is the explicit predictor-

corrector technique Mac-Cormack (Ref 2).

3. The final steady state flow is obtained in the asymptotic

limit of large times. In most cases, this is the desired result.

However, the time-marching procedure can also be used tocalculate the transient behaviour viscous flows.

The numerical solution of the full Navier-Stokes

equations for hypersonic flows is a state of the art research

problem at the present. Many numerical approaches have been

and are being developed and studied, both using explicit and



implicit finite difference method. See Ref 3 for an organised

presentation of such method. A full Navier-Stokes calculation

of the flowfield over a complete 3-D airplane configuration.

Such a calculation has recently been made, for the first time in

the history of aerodynamics, by Joe Shang at the Air-Force

flight dynamic laboratory, and is described in Ref 4. Here, the

viscous flow is calculated over the X-24C hypersonic research

vehicle at . A three view of the X-24C is shown in

figure 1.

Fig. 1`. Three view of the X -24C hypersonic test vehicle

The calculation carried out by Shang has the following

characteristics:

1. The complete Navier-Stokes eqs were used in a

conservation form derivable from Eqs. (1-5)

2. The Baldwin-Lomax turbulence model was employed. See

Ref. 5

3. Mac-Cormack explicit predictor-corrector finite difference

scheme in precisely the same form as described in Ref. 2 wasused for the numerical solution of the Navier-Stokes eqn.

4. The shock capturing approach was taken.

5. A mesh system consisting of 475,200 grid points as

distributed over the flowfield.

Sample results from the calculation are shown in Figure 2-

4. In fig. 2a and b, peripheral surface pressure distribution are

given as a function of normalised arc length at various stream

wise stations denoted by x/ , where is the nose radius. By peripheral distributions, what is meant is a distribution along

a body surface generator that goes from the top to the bottom

of the vehicle at a given stream-wise station; these peripheraldirections are clearly shown in fig. 3 which is a perspective

view of the X-24C. In fig. 2 the normalised arc length is

defined as the length measured from the top of the vehicle

toward the bottom, divided by the total arc length of each

individual cross section. For graphical clarity, each peripheral

Fig. 2a

Fig. 2b

Figure 2 Peripheral surface pressure distributions around the X -24C;

comparison between experiment and the Navier-Stokes calculations of Shang

Fig. 3 Illustration of the peripheral direction around the X-24C for the data

shown





Fig. 7 Velocity Distribution in the nozzle

Fig. 8 Temperature distribution in the nozzle

The figures 5-8 show coherence with the actual

phenomenon generally observed in a converging-diverging

nozzle. The computations of these results took a large amount

of time and hardware capabilities.

IV. CONCLUSION

Here, the complete Navier-Stokes equations are solved bymeans of a time marching approach. This method is the

ultimate in conceptual accuracy allowing for pressure

gradients and flow separation to occur would be the case in the

natural flow problem. Such Navier-Stokes solutions,

especially for three-dimensional flows, although carried out in

practice today, are still state of the art research calculations.

Computer storage requirements and running times can be

enormous for such calculations.

V. REFERENCES

1. Hypersonic and High Temperature Gas Dynamics, John D.

Anderson, Jr. Chap. 6 Eq. 6.1-6.6

2. The Effect of Viscosity in Hypervelocity Impact Cratering,Mac Cormack, R.W. AIAA Paper no. 69-354,19693. Computational Fluid Mechanics and Heat Transfer,

Anderson, D.A., J.C. Tannehill

4. Navier-Stokes Solution for a Complete Re-Entry

Configuration, Shang, vol. 23, no. 12

5. Thin Layer Approximation and Algebraic Model for

Separated Turbulent Flows, Baldwin and H. Lomax, AIAA

Paper no. 78-257.

6. Pressure Tests of the AFFDL X-24C Model at Mach

Numbers of 1.5, 3.0, 5.0 and 6.0, Wannernwetsch, G .D.

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Computational Fluid Dynamic Solutions of Hyper Sonic Viscous Flows