numerical simulation of a supersonic cruise nozzle

7/28/2019 Numerical Simulation of a Supersonic Cruise Nozzle

http://slidepdf.com/reader/full/numerical-simulation-of-a-supersonic-cruise-nozzle 1/9

American Institute of Aeronautics and Astronautics

1

Numerical Simulation of a Supersonic Cruise Nozzle

Balasubramanyam Sasanapuri1

ANSYS Fluent India Pvt Ltd., Pune - 411057, India

Manish Kumar 2, Sutikno Wirogo3 and Konstantin A. Kurbatskii4

ANSYS Inc., Lebanon, NH 03766, USA

The prototyping and testing of a supersonic cruise nozzle that covers wide range of

nozzle geometry configurations (to help with large operating envelop of an aircraft) is both

time consuming and expensive. Numerical simulations offer quick and less expensive

solution to reduce the design time and cost. In the present study one of the configurations of

a supersonic cruise nozzle is simulated for a range of nozzle pressure ratios and the results

are compared with experimental data. The pressure-based and density-based solvers in

ANSYS Fluent CFD code are used for the validation study and solution based adaption is

examined to determine if the accuracy can be improved by local mesh refinement. The

simulation results show very good agreement with the experimental data, and this study

demonstrates an optimized simulation process which can be used to study the entireenvelope of flow and nozzle geometry conditions.

I. Introduction

A supersonic cruise aircraft must be capable of operating over a wide range of altitude and velocity, which

includes subsonic take-off and landing, subsonic cruise, climb and supersonic cruise. One of the solutions for

meeting these varied requirements is a variable-cycle engine, which uses variable-geometry nozzle and combustion

arrangement to operate like a turbofan or turbojet or a hybrid combination to suite the mission requirement. The

design of such a variable geometry nozzle requires testing for a broad envelope of flow conditions and geometry

variations. Prototyping and testing for such an envelope would be very time consuming and expensive.

Computation Fluid Dynamics (numerical simulation) offers a faster and cheaper solution to reduce the design time

and cost. In the present study one of supersonic cruise nozzle configurations is simulated for a range of Nozzle

Pressure Ratio (NPR) values and the results are compared with experimental data. The pressure-based coupled

solver (PBCS) and density-based coupled solver (DBNS) formulations implemented in the general purpose CFD

code (ANSYS Fluent1) are used for the validation study. Solution based adaption is applied to determine if the

accuracy can be improved. Exploratory calculations were done for a free stream Mach number of 0.6 and NPR 2.5,

and the best solution process established from these calculations were then used for a second set of cases with zero

free stream Mach number and a NPR range from 2.5 to 7.0. The problem is described in detail in Sec. II, an

overview of the solver algorithm is given in Sec. III, and finally the numerical predictions are presented and

compared with experimental data in Sec. IV.

II. Problem Description

The problem considered for this study is a 2D axi-symmetric configuration of a supersonic cruise nozzle (Fig. 1).

The geometry and flow conditions correspond to one of the nozzle configurations studied experimentally in Ref. [2].

The study was carried out in two parts: the first part focused on the effects of mesh refinement and different solver

settings to get the most accurate solution in comparison to the experimental data. The second part considered a

parametric study for a series of NPRs using the best settings derived from the first part of the study.

1 Senior Technology Specialist.2 Senior Technical Services Engineer.3 Senior Technical Account Manager.4 Lead Technical Services Engineer, Senior AIAA Member.

51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition07 - 10 January 2013, Grapevine (Dallas/Ft. Worth Region), Texas

AIAA 2013-049

Copyright © 2013 by ANSYS, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.




2

Figure 1. 2D axi-symmetric geometry of a supersonic cruise nozzle.

III. Numerical Model

Most numerical approaches to high speed jet flows, e.g.3-4, employ density-based coupled formulations where the

governing equations of continuity, momentum, energy and (where appropriate) species transport are solved

simultaneously as a set, or vector, of equations. In this approach, density is used as a primary variable found from

the continuity equation, and then pressure is deduced from it using an equation of state. Density-based techniques

are found to be efficient when used for high subsonic, transonic or supersonic flows; however they requiremodifications, such as preconditioning 5-6 in low Mach number flow regions (e.g. stagnation region outside the jet in

the outer domain) to overcome the problem of the system matrix becoming singular in the incompressible limit. The

density-based double-precision implicit solver 1 with preconditioning is used in this work.

As an alternative to the density-based approach, a number of coupled pressure-based methods have been

proposed 7 – 11 to extend applicability of pressure-based segregated techniques to problems where the inter-equation

coupling is strong. Unlike a segregated algorithm, in which the momentum equations and pressure correction

equation are solved one after another in a decoupled manner, a pressure-based coupled algorithm solves a coupled

system of equations comprising the momentum equations and pressure correction equation. Since the momentum

and pressure equations are solved in a closely coupled manner, the rate of solution convergence significantly

improves when compared to a segregated solver. The coupling also makes pressure-based coupled algorithms

applicable to supersonic and hypersonic problems, which can be very difficult to solve by a segregated approach. A

pressure-based coupled double-precision solver 1 is employed in this study to compare with the conventional density-

based solver. The solver algorithms are briefly discussed in the sections below.

A. Density-based Coupled Solver (DBNS)

The governing equations for the conservation of mass, momentum and energy are discretized using a control-

volume-based technique. The system of governing equations for a single-component fluid is cast in integral

Cartesian form for an arbitrary control volume V with differential surface area d A as follows:

V V

dV d dV t

HGFW A][ (1)

where the vectors W, F and G are defined as,

T],,,,1[ E wu v W , T],ˆ,ˆ,ˆ[ vvk v jvivvF p E pw p pu v, ,

T],,,,0[ qG jij zi yi xi v (2)

and the vector H contains source terms such as body forces and energy sources. Here , v, E , and p are the density,velocity, total energy per unit mass, and pressure of the fluid, respectively, is the viscous stress tensor, and q is the

heat flux. Total energy E is related to the total enthalpy H as E = H – p / where H = h + |v|2/2 and h is sensible

enthalpy. The Navier-Stokes equations (1) become numerically very stiff at a low Mach number due to the disparity

between the fluid velocity and the acoustic speed of sound. The numerical stiffness of the equations under these

conditions results in poor convergence rates. This difficulty is overcome by employing time-derivative

preconditioning6, which modifies the time-derivative term in (1) by pre-multiplying it with a preconditioning matrix.

This has the effect of re-scaling the acoustic speed (eigenvalue) of the system of equations being solved in order to

alleviate the numerical stiffness encountered at low Mach numbers and in incompressible flows. Face values




3

required for computing the convection terms are interpolated from the cell centers using a second-order upwind

scheme12.

The inviscid flux vector F appearing in (1) is evaluated by Advection Upstream Splitting Method 13 (AUSM).

ANSYS Fluent utilizes an all-speed AUSM+ scheme14 based on the low Mach number preconditioning. The coupled

set of governing equations (1) is discretized in time using an implicit time-marching algorithm. In the implicit

scheme, an Euler implicit discretization in time is combined with a Newton-type linearization of the fluxes to

produce a linearized system in delta form15. The system is solved using Incomplete Lower Upper (ILU) factorization

in conjunction with an algebraic multigrid (AMG) method 1, 16 adapted for coupled sets of equations. Time marching

proceeds until a steady-state solution is reached. Explicit relaxation is applied to improve the convergence to steady

state by controlling the amount that the solution vector changes between iterations after the end of the AMG cycle.

Gradients needed for constructing values of a scalar at the cell faces and for computing secondary diffusion

terms and velocity derivatives are calculated using the least squares cell-based gradient evaluation 1 which preserves

a second-order spatial accuracy.

B. Pressure-based Coupled Solver (PBCS)

An implicit discretization of the pressure gradient terms in the momentum equations, and an implicit

discretization of the face mass flux, including the Rhie-Chow pressure dissipation terms, provide fully implicit

coupling between the momentum and continuity equations. This discretization yields a system of algebraic equations

whose matrix depends on the discretization coefficients of the momentum equations1, which is then solved using the

coupled algebraic multigrid (AMG) scheme1, 16. An ILU smoother is applied to smooth the residuals between levels

of the AMG. The ILU smoother is more expensive than standard Gauss-Seidel, but has better smoothing properties,especially for block-coupled systems solved by the coupled AMG, which permits more aggressive coarsening of

AMG levels.

Either a second-order upwind scheme12 or QUICK-type scheme1, 17 scheme is used for interpolating face values

of velocities and energy. The QUICK scheme implementation in ANSYS Fluent is based on a weighted average of

second-order-upwind and second-order central differencing of the variable. It uses a variable, solution-dependent

value of the weight factor, chosen so as to avoid introducing new solution extrema.

Face values of pressure are reconstructed using a second-order or PRESTO! scheme. The second-order

implementation is similar to a multidimensional linear reconstruction approach12. In this approach, higher-order

accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell

centroid. The PRESTO! (Pressure Staggering Option) scheme1 uses the discrete continuity balance for a “staggered”

control volume about the face to compute the “staggered” (i.e., face) pressure. This procedure is s imilar in spirit to

the staggered-grid schemes used with structured meshes18. For triangular, tetrahedral, hybrid, and polyhedral

meshes, comparable accuracy is obtained using a similar algorithm.

C. Gradient Limiters

Both DBNS and PBCS solver formulations take advantage of gradient (or slope) limiters used on the second-

order upwind scheme to prevent spurious oscillations, which would otherwise appear in the solution flow field near

shocks, discontinuities, or near rapid local changes in the flow field. The gradient limiter attempts to invoke and

enforce the monotonicity principle by prohibiting the linearly reconstructed field variable on the cell faces to exceed

the maximum or minimum values of the neighboring cells. A non-differentiable limiter 12 based on the Minmod

function (Minimum Modulus) is utilized in this study to limit and clip the reconstructed solution overshoots and

undershoots. Cell to face limiting direction is chosen, where the limited value of the reconstruction gradient is

determined at cell face centers.

D. Physical Models and Boundary Conditions

Air is modeled as a single-species ideal gas. For the NPRs considered in this study, the maximum Mach number is expected to be below 4.0, hence real-gas thermodynamic non-equilibrium processes are not expected to have a

strong effect on aerodynamic heating, and therefore an aerothermochemical model is not taken into account in the

simulation. The numerical code ANSYS FLUENT provides capabilities to include chemical and vibrational non-

equilibrium effects when they cannot be neglected. Paterna et al19 discussed one such example application of the

code, with chemical non-equilibrium enabled, to the Martian atmosphere entry problem.

The effects of turbulence are modeled using the shear-stress transport SST k-model proposed by Menter 20. It

effectively blends the robust and accurate formulation of the k-model21 in the near-wall region with the free-

stream independence of the k- model22 in the far field. To achieve this, the k- model is converted into a k-




4

formulation. The standard k- model and the transformed k- model are both multiplied by a blending function, and

both models are added together. The blending function is one in the near-wall region, which activates the standard k-

model, and zero away from the surface, which activates the transformed k- model.

The pressure inlet boundary condition at the nozzle inlet specifies static and total pressure, total temperature and

flow direction which correspond to the test conditions2. Boundary values of turbulent kinetic energy and its specific

dissipation rate at the nozzle inlet are derived from the turbulence intensity I = 5% and the nozzle inlet diameter D

as,

where U i is calculated inlet velocity and C = 0.09 is the model constant.

The far-field boundary is treated as a pressure far-field where the free stream Mach number, static pressure and

static temperature are specified. This non-reflecting boundary condition is based on the introduction of Riemann

invariants (i.e., characteristic variables) for a one-dimensional flow normal to the boundary.

The downstream outlet boundary is treated as a pressure boundary which fixes specified static pressure and

extrapolates all other flow variables from the interior of the domain if the flow is locally subsonic. In supersonic

regions no boundary conditions are

imposed, and all flow variables

(including static pressure) are

extrapolated from the interior. A no-

slip condition is set for all the walls,and the walls are treated as adiabatic

surfaces.

E. Baseline Computational MeshA 2D axi-symmetric domain

containing quadrilateral mesh (Fig. 2)

is generated using ANSYS Pre-

processing tools1. The mesh size

distribution is determined by multiple

considerations, including the need to

accurately capture shock structures in

the divergent portion of the nozzle, and

shock induced separation. The mesharound all the nozzle wall surfaces

contains a boundary layer type mesh

fine enough to resolve the viscosity-

affected near-wall region all the way to

the laminar sublayer to ensure y+ in the wall-adjacent cell is on the order of one. The total mesh size is 359 thousand

cells.

IV. Numerical Results and Comparison with Test Data

The numerical solution is initialized from the free-stream flowfield, and then the full multigrid (FMG)

initialization1 is utilized to obtain the initial solution. The FMG initialization is based on the full-approximation

storage (FAS) multigrid algorithm1, 23. The FMG procedure constructs several grid levels to combine groups of cells

on the finer grid to form coarse grid cells. FAS multigrid cycle is applied on each level until a given order of

residual reduction is obtained, then the solution is interpolated to the next finer grid level, and the FAS cycle isrepeated again from the current level all the way down to the coarsest level. This process is continued until the finest

grid level is reached. FMG initialization is relatively inexpensive since most of computational work is done on

coarse levels, which allows one to obtain a good initial solution that already recovers some flow physics. Once the

initialization is complete, the solver is iterated until a steady-state solution is reached.

A. Grid Independence StudyA solution is initially converged on the baseline mesh, and then the baseline mesh is adaptively refined (referred

to as “adaption”) to increase resolution across the shocks, and to provide a case matrix for the grid independence

Figure 2. Computational mesh.




5

Figure 4. Mach number contours

(baseline mesh)

study. Two levels of adaption are applied to capture the shock diamond features in the divergent portion of the

nozzle and the downstream (Fig. 3). The adaptive refinement is based on the Blast Wave Identification Parameter

(BWIP) approach24 modified for steady-state flows. This adaption method follows the observation that the Mach

number normal to the shock passes through the value of one at the shock. A BWIP for tracking shock locations can

be constructed using the component of the Mach vector normal to the shock. The orientation of the shock is

determined by the pressure gradient, which is always normal to the shock. The BWIP is a real function defined as

the dot product of the Mach number vector with the pressure gradient as following,

||/ p p M f BWIP

(3)

Figure 3. Computational meshes included in the grid independence study. (a) baseline mesh before adaption,

and (b) mesh after two levels of adaption

The shock position is identified by an iso-surface formed by

grid locations where BWIP f is equal to one. Because the dot

product of the Mach vector and pressure gradient is less than zero

in areas of expansion, the BWIP formulation (3) excludes

expansions. This is intentional for the present application as only

the shocks are of interest. If expansions were to be tracked, then a

second indicator would be used where BWIP f evaluates to

negative one.

A first level of mesh adaption is applied to refine the baseline

mesh using the converged baseline solution. Then, a second level

of adaption is utilized to refine the first level adaption mesh

further. Several simulations were done using PBCS and DBNS

solvers, as discussed in the next section.

B. Results and Comparison with Experimental Data

The first part of the study was carried out for free stream Mach

number of 0.6 and NPR of 2.5. A total of seven runs were

performed: one run each with the PBCS and second order

discretization schemes; PBCS, PRESTO! discretization for

pressure and QUICK scheme for momentum and energy; DBNS

with the second order discretizations and AUSM+ flux splittingmethod. The same combination was repeated with one level of

adaption using the BWIP method discussed above. The seventh

run was performed with PBCS, PRESTO, QUICK combination by

performing a second level of adaption using BWIP.

Figure 4 shows the Mach number contours from different runs

on the baseline mesh, and Fig. 5 plots Mach number contours

before and after the mesh adaption (DBNS solution). Numerical predictions of the pressure distribution along the

internal nozzle wall are compared with experimental data in Fig. 6. All the numerical curves from PBCS and DBNS

(a) (b)




6

solutions in Fig. 6 are nearly identical, and they match favorably with the experimental data. The results from the

first part of the study are summarized in Table 1. Both PBCS and DBNS solvers show excellent agreement of the

discharge coefficient with the experimental data, whereas DBNS solver shows the best match with the experimental

data for the thrust parameter. Shock diamonds are captured quite well by both solvers (Fig. 4). The DBNS with the

second order upwind discretization and PBCS with PRESTO! and QUICK schemes offer better resolution than the

PBCS with the second order upwind discretization. The solution adaptation did not show noticeable improvement in

the C d

or Cfg values indicating the mesh resolution mesh is already sufficient enough to capture the shock diamonds.

Based on the observations from this part of the study, the DBNS solver was used for the Part 2, where the free

stream Mach number is zero and NPRs studied are 2.5, 4, 5, 6 and 7. Contours of Mach number and static pressure

are shown in Fig. 7 and 8, respectively. Comparison with

experimental data is presented in Fig. 9 and Table 2.

Calculated C d and Cfg values compare very well with the

experimental data over a broad range of NPRs, where the

maximum error for C d is below 0.52% and for Cfg it is

below 2.9%, which is excellent for this class of the flows.

C d

(Exp)

C d

(CFD)

C d ,

error %

Cfg

(Exp)

Cf g

(CFD)

Cfg ,

error %

PBCS: 2 nd order 0.97 0.96513 -0.502 0.71 0.67547 -4.863

PBCS: 2 nd order, adapted 0.97 0.96530 -0.485 0.71 0.67650 -4.706

PBCS: PRESTO, QUI CK 0.97 0.96510 -0.505 0.71 0.68475 -3.556

PBCS: PRESTO, QUI CK, adapted 0.97 0.96482 -0.534 0.71 0.68504 -3.515

PBCS: PRESTO, QUI CK, adapted twice 0.97 0.96510 0.506 0.71 0.6857 -3.422

DBNS 0.97 0.96482 -0.534 0.71 0.69442 -2.194

DBNS: adapted 0.97 0.96488 -0.528 0.71 0.69446 -2.189

Table 1. Comparison of Discharge Coefficient (C d ) and Thrust Coefficient (Cfg ) for the first part of the study.

Free stream Mach = 0.6 and NPR = 2.5.

Figure 5. Mach number contours (effect of

adaption)

Figure 6. Pressure distribution along the internal

nozzle wall. p t,j – jet total pressure, and p amb – free-

stream static pressure.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

p t , j

/ p a m b

x / dm

PBCS: 2nd order

PBCS: 2nd order adapted

PBCS: PRESTO, QUICK

PBCS: PRESTO, QUICK adapted

DBNS

DBNS: adapted

Experimental




7

Figure 7. Mach number contours for different NPRs.

Figure 8. Pressure contours for different NPRs.




8

Figure 9. Comparison of C d and Cfg with experimental data for different NPRs.

NPR C d (Exp) C d (CFD) C d , error % Cfg (Exp) Cf g (CFD) Cfg , error %

2.5 0.970 0.96496 -0.519 0.790 0.76925 -2.626

4.0 0.970 0.96510 -0.505 0.873 0.84793 -2.872

5.0 0.970 0.96520 -0.496 0.889 0.87535 -1.536

6.0 0.968 0.96525 -0.284 0.906 0.90276 -0.358

7.0 0.967 0.96530 -0.176 0.923 0.92846 0.590

Table 2. Comparison of C d and Cfg with experimental data for different NPRs.

V. Conclusion

This study confirms the ability of a general purpose CFD solver using both density-based and pressure-based

algorithms to accurately resolve the complex physics of a supersonic nozzle flow. The numerical predictions of

nozzle coefficients and surface pressures inside the nozzle are in excellent agreement with experimental data. A

grid-independence study is carried out to provide an additional verification of the CFD results. The BWIP adaption

approach proves to be very effective in identifying and resolving shocks of disparate strengths.

The results reported in this work show that the pressure-based coupled solver (PBCS) algorithm can be used as a

robust and effective method that can adequately resolve the physics and capture all essential features of supersonicflows. PBCS is less memory and CPU intensive than a traditional density-based approach, which makes it an

economically attractive alternative to the density-based formulations in simulating supersonic nozzle problems.

References1 ANSYS, Inc. Product Documentation Release 14.5, ANSYS, Inc., Canonsburg, PA 15317, USA, 2012. 2 Carson, G. T. Jr., and Lee, E. E. Jr., “Experimental and Analytical Investigation of Axisymmetri Supersonic Cruise Nozzle

Geometry at Mach Numbers from 0.60 to 1.30,” NASA TP 1953, 1981.3 Tamada, I., Aso, S., and Tani, Y., “Numerical Study of the Effect of the Opposing Jet on Reduction of Aerodynamic

Heating with Different Nose Configurations,” AIAA Paper 2005-188, 2005.4 Gnemmi, P., Srulijes, J., and Roussel, K., “Flowfield Around Spike-Tipped Bodies for High Attack Angles at Mach 4.5,”

AIAA Journal of Spacecraft and Rockets, Vol. 40, No. 5, 2003, pp. 622-630.5 Venkateswaran, S., Weiss, J. M., and Merkle, C. L., “Propulsion Related Flowfields Using the Preconditioned Navier-

Stokes Equations,” AIAA paper 92-3437 .6 Weiss, J. M. and Smith, W. A., “Preconditioning Applied to Variable and Constant Density Flows,” AIAA Journal , Vol. 33, No. 11, 1995, pp. 2050-2057.

7 Vanka, S. P., “A Calculation Procedure for Three-dimensional Steady Recirculating Flows Using Multigrid Methods,”Computer Methods in Applied Mechanics and Engineering , Vol. 55, 1986, pp. 321 – 338.

8 Jyotsna, R., and Vanka, S. P., “Multigrid Calculation of Steady, Viscous Flow in a Triangular Cavity,” Journal of Computational Physics, Vol. 122, 1995, pp. 107 – 117.

9 Raw, M., “Robustness of Coupled Algebraic Multigrid for the Navier -Stokes Equations,” AIAA Paper 96-0297 .10 Smith, K. M., Cope, W. K., and Vanka, S. P., “A Multigrid P rocedure for Three-dimensional Flows on Non-orthogonal

Collocated Grids,” International Journal for Numerical Methods in Fluids, Vol. 17, 1993, pp. 887 – 904.

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7

C f g

NPR

Experiment

CFD0.90

0.92

0.94

0.96

0.98

1.00

1 2 3 4 5 6 7

C d

NPR

Experiment

CFD




9

11 Webster, R., “An Algebraic Multigrid Solver for Navier -Stokes Problems,” International Journal for Numerical Methods in Fluids, Vol. 18, 1994, pp. 761 – 780.

12 Barth, T. J., and D. Jespersen, D., “The Design and Application of Upwind Schemes on Unstructured Meshes,” Technical

Report AIAA-89-0366 , AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, 1989.13 Liou, M. S., and Steffen, C. J., Jr. , “A New Flux Splitting Scheme,” Journal of Computational Physics, Vol. 107(1), 1993,

pp. 23-39.14 Liou, M. S., “A Sequel to AUSM: AUSM+,” Journal of Computational Physics, Vol. 129, 1996, pp. 364-382.15

Weiss, J. M., Maruszewski, J. P., and Smith, W. A., “Implicit Solution of the Navier-Stokes Equations on UnstructuredMeshes,” Technical Report AIAA-97-2103, 13th AIAA CFD Conference, Snowmass, CO, 1997.16 Hutchinson, B. R., and Raithby, G. D., “A Multigrid Method Based on the Additive Correction Strategy,” Numerical Heat

Transfer , Vol. 9, 1986, pp. 511-537.17 Leonard, B. P., and Mokhtari, S., “ULTRA-SHARP Nonoscillatory Convection Schemes for High-Speed Steady

Multidimensional Flow,” NASA TM 1-2568 (ICOMP-90-12), NASA Lewis Research Center, 1990.18 Patankar, S. V., Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington, DC. 1980.19 Paterna, D., Monti, R., Savino, R., and Esposito, A., “Experimental and Numerical Investigation of Martian Atmosphere

Entry,” AIAA Journal of Spacecraft and Rockets, Vol. 39, No. 2, 2002, pp. 227-236.20 Menter, F. R., “Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications,” AIAA Journal , Vol. 32,

No. 8, 1994, pp. 1598-1605.21 Wilcox, D. C., Turbulence Modeling for CFD, DCW Industries, Inc., La Canada, California, 1998.22 Launder, B. E., and Spalding, D. B., Lectures in Mathematical Models of Turbulence, Academic Press, London, England,

1972.23 Brandt, A., “Multi-level Adaptive Computations in Fluid Dynamics,” AIAA Paper 79-1455.24 Kurbatskii, K. A., Montanari, F., Cler, D. L. and Doxbeck , M., “ Numerical Blast Wave Identification and Tracking Using

Solution-Based Mesh Adaptation Approach,” AIAA Paper 2007-4188.

numerical simulation of a supersonic cruise nozzle

Documents