Parallel Computational Fluid Dynamics '95

Parallel Computational Fluid Dynamics:

Implementations and Results Using Parallel Computers

A. Ecer, J. Periaux, N. Satofuka and S. Taylor (Editors)

9 1995 Elsevier Science B.V. All rights reserved.

97

Numerical Simulation of Reacting Mixing Layer with Combined Parallel

Approach

Edgard Kessy, Alexei Stoukov and Dany Vandromme ~

~LMFN, INSA of Rouen, URA-CNRS 230-CORIA, 76821 Mont Saint Aignan CEDEX,

France

This work concerns the parallelization of an explicit algorithm for the simulation of

compressible reacting gas flows, applied to supersonic mixing layers.

The reactive Navier-Stokes equations are characterized by three tightly coupled physical

phenomena, i.e. the convection, diffusion and chemical source terms. To compute the

chemical source terms, full complex chemistry is used. By considering the elapsed time

for solving the problem, the numerical treatment of the chemical source terms takes about

75% of the total execution time.

The main goal of the present work is to reduce the relative cost of chemical source

terms calculation and also to optimize the global cost of the procedure resolution by the

use of parallel computation.

1. Choice of Parallel Approach

On parallel architectures, computations are executed simultaneously on several pro-

cessors. To achieve this goal, the algorithm is divided into several independent tasks.

All tasks can be executed simultaneously and communicate with each other during the

execution.

Two different types of parallel methodologies exist: data-parallelism and control paral-

lelism [14]. The first approach relies on the fact that each processor performs at a given

time the same instruction on different data. This approach exploits the great potential

of massively parallel computers, SIMD (Single Instruction Multiple Data) architectures

[11].

In the control-parallelism approach, the computational problem is divided into a number

of independent tasks, with different processors performing different tasks in parallel. This

approach is adapted to multi-processor computers, MIMD (Multiple Instruction Multiple

Data) architectures [1].

In order to use MIMD computers very efficiently, the granularity of tasks should be as

large as possible. In CFD the decomposition of the computational domain is the most

efficient technique in order to increase the granularity of tasks for MIMD architectures

[5]. This technique (the so-called multi-domain technique) consists in partitioning the

computational domain into a number of blocks, and to distribute blocks onto different

processors [4]. However, there is still a problem to maintain a well balanced decomposition.

The choice of the adopted parallel approach is influenced by the numerical algorithm.

98

However, one can notice that data-parallel and control-parallel approaches can both be

combined [12]. By considering that in most reacting flows, reacting and non-reacting zones

occur simultaneously, the computation of chemical source terms can be restricted to the

reactive region. Thus a decomposition efficient for pure hydro dynamical problem becomes

inefficient when the reacting zone dimensions differ greatly between blocks. In such a way,

the standard multi-block technique is no longer well suited for the reacting flow. In this

paper, a multi-block technique is used for convective and diffusive terms, whereas SPMD

(Single Program Multiple Data) or SIMD approach is employed for chemical source terms.

Figure 1. Flow configuration

2. Flow configuration, physical, chemical and mathematical model

Figure 1 shows the physical model considered in the present study. It consists of two

chemically active hydrogen and air streams with different stream wise velocities. The

spatial mixing of reacting streams has been simulated in a two-dimensional domain.

The static pressure at the inlet side is the same

for both streams. To prescribe the inlet condi-

tions the self similar solution of the compress-

ible mixing layer [9] is used at the inlet with

a vorticity thickness equal to 0.05 of the trans-

verse length. The inlet conditions are presented

in Table 1.

Table 1

Stream Velocity Mach Pressure Temperature Convective

m/s

number

Pa K

Mach number

/-/2 3000 1.24 1.013-l0 s 930- 1000 0.446

AIR

1780 2.803

The flow evolution is governed by the unsteady compressible Navier-Stokes equations

coupled with the energy and species transport equations. These equations are written in

two-dimensional form as:

OU OF(U) OG(U) = S(U)

(1)

O----t -~ Ox t Oy

where the variable vector, the convective and diffusive fluxes and source terms are defined

respectively by:

U = [Pl,""",

Pnsp, pu, pv, pet] T

F=Fc+Fv G=Gc+Gv

Fc

= [plu,"',

PnspU, pUU + P, puv, (pet + p)u] T

[

Fv = - ~rylx, . . . , ~ryn~.px , ~rx~, cr~y, cr~u + axvv - q~

Gc

= [Plv,"',

PnspV, pVU, pVV -t- p, (pet + p)v] T