Multiple Pressure Variable (MPV) Approach for

Multiple Pressure Variable (MPV) Approach for Low Mach Number Flows Based on Asymptotic Analysis K.J. Geratz, R. Klein RWTH Aachen Institut fur Technische Mechanik Templergraben 64, 52056 Aachen, Germany C.D. Munz, S. Roller Forschungszentrum Karlsruhe, Technik und Umwelt Institut fur Neutronenphysik und Reaktortechnik PO-Box 3640 76021 Karlsruhe, Germany SUMMARY An asymptotic analysis of the compressible Euler equations in the limit of vanishing Mach numbers is used as a guideline for the development of a low Mach number extension of an explicit higher order shock capturing scheme. For moderate and large Mach numbers the underlying explicit compressible ow solver is active without modi cation. For low Mach numbers, the scheme employs an operator splitting technique motivated by the asymptotic analysis. Advection of mass and momentum as well as long wave acoustics are discretized explicitly, while in solving the sonic terms, the scheme uses an implicit pressure correction formulation to guarantee both divergence-free ow in the zero Mach number limit and appropriate representation of weakly nonlinear acoustic eects for small but nite Mach numbers. This asymptotics based approach is also used to show how to modify incompressible ow solvers to capture weakly compressible ows.

1. INTRODUCTION 1.1 Motivation One motivation of the work to be presented stems from combustion applications. Reliable simulation of slow ames, ame acceleration and the de agration-to-detonation transition (DDT) require an "all Mach number capability" of the simulation code: Laminar premixed combustion generates ows with Mach numbers Mlam = 10?4 : : : 10?3, while a

detonation wave attains lead shock Mach numbers Mdeto = 5 : : : 10. A considerable variety of numerical techniques for zero and/or low Mach number and weakly compressible ows have been proposed in recent years, e.g., [1],[3],[4],[5],[6],[7]. Nevertheless, each of these schemes fails to satisfy at least one of the following necessary requirements, which are motivated by the de agration-to-detonation transition phenomenon. The scheme should 1. yield an acceptable numerical approximation for the equations of zero Mach number

ow with large amplitude density variations [12], as M = 0, 2. provide an accurate representation of the long wave acoustics responsible for ow acceleration to higher Mach numbers, and it should 3. reduce to a reasonable explicit full compressible Euler solver for characteristic ow Mach numbers of order M  1 and larger. The principal idea behind the new class of schemes is to rst perform an asymptotic analysis for small Mach numbers and to use the insight gained for designing a suitable numerical discretisation [8],[9],[10],[11]. It needs to be emphasized, that this procedure yields a scheme for the full Euler equations, not just one for solving some reduced asymptotic system. In this sense, the asymptotic analysis is used here in a quite unusual fashion: Rather than providing simpli ed limit equations, which are then solved by highly specialized methods with restricted applicability, the asymptotics serve as a guideline in designing numerical methods for the full equations, which operate eciently even in a certain singular limit regime.

1.2 Reactive Euler equations The compressible reactive Euler equations in nondimensional form read as

t m ~t et (Y )t

+ r
m~ + r
(m ~  ~v + M1 pI ) + r
(~v[e + p]) + r
(m~ Y ) 2

= 0 = 0 = 0 = ?!

(1)

where , m ~ , e and Y are the conserved quantities mass, momentum, total energy per unit volume and the mass fraction of the unburned species, ~v  m ~ = is the ow velocity, p the pressure, ! the local reaction rate, I denotes the unit tensor and the '' symbol indicates the tensorial product. A premixed fuel is assumed with only two species present, unburned and burned, whose thermodynamics are governed by the same -gas law and have the same molecular weights. Furthermore, the ratio of speci c heats is given by

= cp =cv = constant, and the heat release per unit mass of unburned gas by q0 . The pressure is then related to the conserved quantities by the equation of state for mixtures of perfect gases,

~ p = ( ? 1) e ? M 2 m "

2

2 ? Y q0

#

with

= const:

(2)

The reference values used, are density ref , velocity vref , length lref , species Yref and an independent reference for the pressure pref . The reference velocity vref is then independent of cref = (pref =ref )1=2 and the nondimensional velocities ~v and ~c remain well-de ned and

nite in the limit of vanishing Mach number M = q vref ! 0 : pref =ref

(3)

The parameter M is called the global Mach number and characterizes the nondimensionalization but not the local ow Mach number. The incompressible ow is then clearly distinguished from the compressible ow by the vanishing global Mach number (M=0). The wave speed vectors of the nondimensional Euler equations (1) are ~a1 (~n) = ~v + M1 c~n; ~a2 = ~v : (4) The ~a1 (~n) wave speed vectors become arbitrarily large as M converges to zero. As a consequence, the hyperbolic Euler equations converge towards a hyperbolic/elliptic system with in nite propagation rates of perturbations. Theoretical insight into this singular limit is obtained by a singular perturbation analysis for M ! 0.

2. SUMMARY OF ASYMPTOTIC ANALYSIS 2.1 Expansion ansatz In a de agration-to-detonation process the ow acceleration from low to high Mach numbers is caused by the action of acoustic pulses, that are generated by the unsteady premixed turbulent ame. All ow phenomena are then dominated by the time scale of the propagation of the unsteady ame front. The relevant asymptotic regime involves a single time scale (t), but multiple length scales represented by dierently scaled spacial coordinates (~x; ~). The small scale variable ~x resolves entropy uctuations and vortex structures, while the large scale variable ~ resolves acoustic pressure changes. The acoustic waves as well as the small scale incompressible ow both have a leading order in uence on the velocity eld. An appropriate asymptotic expansion for the terms in the Euler equations, (e.g. density, velocity, pressure, mass fraction total energy per volume and reaction rate), U = (;~v; p; Y; e; !) reads as U (~x; ~; t) = U (0) (~x; ~; t) + MU (1) (~x; ~; t) + M 2 U (2)(~x; ~; t) +


; M  1; (5) with ~ = M~x. This expansion is introduced into the compressible reactive Euler equations (1) and the equation of state (2) by using the relation r = r~x + M r~ for spatial derivatives. Using large scale dierencing and spatial averaging, the following results are obtained (see [8] for details).

2.2 Pressure expansion The pressure expansion exhibits three distinct physical roles of "the pressure" in a single time - multiple space scale regime: 1. The leading order pressure p(0) is spatially homogeneous and plays the role of a thermodynamic variable, i.e. p(0) = P0(t). It is also called mean or background pressure. 2. The rst order pressure p(1) varies spatially only on the acoustic length scale, i.e., p(1) = P (1) (~; t), and re ects the in uence of long wave acoustics [8]. Flows that undergo considerable changes of the characteristic ow Mach number involve acoustic pressure amplitudes, which produce leading order eects on the velocity eld. Therefore, the term Mp(1) in the pressure expansion must be considered and may not be excluded, as usually done in asymptotic derivations of the incompressible limit [12]. 3. The second order pressure p(2) has small scale and acoustic scale spatial structure, i.e., p(2) = p(2) (~x; ~; t). It acts as a local ballance of forces agent and guarantees the divergence constraint of incompressible ows as M ! 0. Hence, the pressure expansion scheme reads as p = P0(t) + MP (1) (~; t) + M 2 p(2) (~x; ~; t) + O(M 3 ): (6)

2.3 Short wavelength dynamics On the small convective length scale, which is identical to the reference length lref , the leading order continuity, the second order momentum and the leading order energy conservation equations yield the following system

(0) + r~x
(~v)(0) = 0 t (0) (0) (2) (~v)t + r~x
(~v  ~v) + r~xp = ?r~P (1) r~x
~v(0) = ? P1 dPdt +

P?1 q0 !(0) 0

0

0

;

(7)

where r~x and r~ refer to the gradient concerning the ~x-scale and ~-scale, respectively. These are the inviscid variable density ow equations for zero Mach number, supplemented by a long wave acoustic pressure gradient source term in the momentum equation (7)2 and global compression as well as thermal expansion source terms in (7)3. The small scale average (i.e., with respect to ~x) of the divergence constraint (7)3 over a domain yields an evolution equation for the background pressure P0 dP0 = ( ? 1)q0 Z !(0)@ ? P0 Z ~v(0)
~nds : (8) dt V ol( )

V ol( ) @

The Gauss-Green theorem has been applied to the ~v(0) -term, where ~n denotes the outward directed unit normal vector on the boundary @ of the domain with volume Vol( ). Two cases have to be distinguished now: In the limit, when Vol( ) becomes large and ~v(0) is bounded, the second term on the right hand side of (8) vanishes. The mean pressure P0 is then determined by the thermal source term. In the case that there is no thermal source term present, the incompressible divergence constraint r~x
~v(0) = 0 is obtained and P0 is constant in space and time. The ow is then divergence-free, except for termal expansion

due to chemical reactions. Another situation occurs when the domain of the uid ow is bounded. The second term on the right hand side of (8) then does not vanish, but rather describes an overall pressure rise due to compression from the boundary. The velocity ~v(0) in a non-reeactive ow is then not necessarily divergence-free and the right hand side of (7)3 can be viewed as a distributed volume source, where a positive contribution results in an expansion of the gas and P0 changes with time. It is interesting to note, that the divergence constraint (7)3 results from the energy equation and not from the continuity equation as usually in the derivation of the incompressible Euler equations. Depending on the overall system dimensions, only one of the supplementary pressure source terms can be present: Systems with dimensions L of order O(lref ) are in the single length scale regime and no long wave acoustics can be accomodated and hence r~P (1)  0. The global compression term in (7)3 is then readily determined by the boundary conditions of the system and the thermal source term. Systems with dimensions L = O(lref =M ) are large enough to accomodate long wave acoustics and therefore lie in the multiple length scale regime. With no chemical source term present, the source term of the divergence constraint vanishes, dP0=dt = r~x
~v  0, while r~P (1) 6= 0. Leading order pressure changes then occur only due to accumulation of rst order pressure wave eects on time scales O(lref =Mvref ) and the local ow divergence is O(M) only. The second order pressure gradient, r~xp(2) , adjusts to guarantee satisfaction of the small scale divergence constraint, (7)3 . Thus, the second order pressure in a low Mach number expansion attains the role of "the pressure" of the incompressible ow equations as M ! 0, [12], [8].

2.4 Long wave length dynamics The ~x-scale-averages of the second order momentum equation (7)2 and the rst order energy equation yield a system, that describes the long wave acoustic momentum exchange: m ~ t + r~P (1) = 0 R ; (9) Pt(1) + r~
(c2 m ~ ) = r~
(c2 ~v~) + (V ?ol1)( )q !(1)@

0



~ ; v denote the ~x-averaged leading order denwith ~ = (0) ? ; v~ = v(0) ? v. Here ; m sity, momentum and velocity. The square of the averaged sound speed is denoted by c2 = P0=. This is the system of linearized acoustics except for the inhomogeneous source terms in (9)2 . The rst source term of (9)2 is generated by the ~x-scale uctuation correlation of density and velocity of multidimensional ows and introduces nontrivial coupling between the small-scale, quasi-incompressible ow and the superimposed long wave acoustics. The second source term describes the in uence of acoustics on the chemical reaction rate. Acoustic waves, emitted by a turbulent ame on the ~x-scale, can lead to autoignition of premixed reactive gases by accumulation of acoustic pressure wave eects and their in uence on the reaction rate. For one-dimensional ows, the acoustic subsystem decouples from the small scale quasi-incompressible ow and there is no feedback from the small scale ow to the long wave acoustics.

3. NUMERICAL METHODS 3.1 Operator Splitting The compressible hyperbolic Euler equations degenerate as M ! 0 and the incompressible limit is approached. The convection terms for mass and momentum remain nonsingular but the divergence constraint (7)3 leads to an elliptic Poisson-type equation for p(2) , while the acoustic system (9) predicts in nitely fast signal propagation in the limit M = 0. This justi es the following numerical operator splitting of the Euler equations (1): System I describes the advection of mass, momentum and species and their changes due to ow divergence,

t + r~x
(m ~) = 0 m ~ t + r~x
(~v  m ~) = 0 (Y )t + r~x
(m ~ Y ) = ?!

9 > = > ;

System I:

(10)

System II describes the ow acceleration due to pressure forces and energy conservation,

m ~ t + M1 r~x p = 0 et + r~x
(H m ~) = 0 2

)

System II;

(11)

where H = (e + p)= and p = ( ? 1)(e ? M 2 m ~ 2 =2 ? Y q0).

3.2 Discretization of System I System I (in one space dimension) is nonstrictly hyperbolic and has a double eigenvalue and only one eigenvector. The term r~xp of order O(M 2 ) is therefore introduced into the momentum equation (10)2 in order to regularize System I.

t + r~x
(m~ ) = 0 m ~ t + r~x
(~v  m ~ + p) = 0 (Y )t + r~x
(m ~ Y ) = ?!

9 > =

:

> ;

System I

(12)

The new System I (12) is strictly hyperbolic and is solved by use of a higher order compressible ow solver with Strang-type splitting [14]. The error, which is introduced by the regularization, is compensated for in System II (13)1 up to second order by subtracting r~xp from (11)1. The pressure and density of System I are coupled by assuming a locally isentropic relationship for density and pressure. Embedding a Godunov-type compressible ow solver into the numerical scheme for System I guarantees, that the scheme operates reliably for low and high Mach number ows including strong shocks.

3.3 Discretization of System II System II is of elliptic/hyperbolic character and contains three distinct physical mechanisms. By introducing the asymptotic expansions for the pressure, the equation of state

and the species the following System II is obtained: 



?r~P (1) m ~ t + r~x p(2) ? p = System II; (13) P P (2) 2 M et + r~x
(~v[e + p]) = ? ?1 ?M ?1 ?(Y q0 )t with r~x( M1 p ? p) = r~x(p(2) ? p) + r~P (1) : The time derivative of the total energy was transformed by the relation 9 =

0t

(1)

t

;

2

(1) e = P?0 1 + M P? 1 + M 2 e(2) + Y q0 :

(14)

In (13), the distinct physical meanings of "the pressure" show up clearly: 1. The pressure p(2) accounts for the local ballance of forces and satis es the divergence constraint of incompressible ows. 2. In the single length scale regimes a global background compression due to changes in the system volume or chemical reactions is introduced by the P0-term. 3. In the multiple length scale regime the propagation of acoustic pressure changes with correct signal speeds is included. The P (1) -terms of the right hand side are obtained from the wave equations (9). System II is solved in a predictor-corrector fashion. Therefore, the changes in momentum and energy are split up into explicit hyperbolic and implicit elliptic contributions. The explicit predictor step then yields either the leading order pressure changes of P0 due to a global background compression, or the dominant rst order O(M) pressure changes of P (1) through a non-dissipative discrete solution of the long wave acoustic system (9). The implicit corrector step then computes the small scale O(M 2 ) pressure uctuations and thereby guarantees compliance with the small scale divergence constraint as M ! 0. Furthermore, the corrector step produces the weakly nonlinear acoustic eects which, for small but nonzero Mach numbers, are neglected in the linear acoustic predictor.

3.4 Predictor step The predictor step determines preliminary updates for momentum and energy in acoustic and chemical predictors. The prediction involves

~ and of approximations to the (i) the determination of small scale averaged momenta m asymptotic rst order pressure P (1) by suitable averaging and summation procedures, (ii) the solution of the linearized acoustic equations (9), and (iii) for more than one space dimension, the inclusion of the coupling term on the right hand side of (9)2 . (iv) In reactive ows, the in uence of the heat release is evaluated. The results of the acoustic predictor are preliminary acoustic updates t m and act e = Mt P (1) =( ? 1) of momentum and energy. They are then employed subsequently in the nal correction step. The chemical source term in (13)2 is treated explicitly as well and contributes to the preliminary energy update by t e = act e + cht (Y q0 ).

3.5 Acoustics on a coarse grid In solving the linearized acoustic system (9), two strategies can be followed: Either, a:, the large scale derivative r~ can be replaced by r~x =M . It is then necessary to apply an implicit scheme in order to comply with the CFL condition or to use an explicit large time step method, [23].
t max  c   1: (15)
x M

Or, b:, the acoustics are solved on a coarse grid with space increments
 =
x=M . The CFL condition then reads as
t max  c  =
t
M max  c   1 ; (16)
 M
x M where the Mach number cancels out. System (9) can then be solved explicitly in an ecient way by the following algorithm:

~ ; c2; P (1) to the coarse grid. 1. Restrict m 2. Solve (9) by an explicit approximation. 3. Prolongate the obtained values for m ~ ; P (1) back to the ne grid by using an appropriate interpolation. In cases with periodic boundary conditions, trigonometric interpolation proved to be superior to linear and cubic interpolation, but all were stable. Small scale disturbances may be introduced by the averaging procedure or by interpolation to the ne grid, and may cause small oscillations in the numerical solution of the acoustic equations. But the solution only serves as prediction of the long wave eects and will be compensated for in the corrector step.

3.6 Corrector step ~ and t e, the corrector solves for the Given the linear acoustic and chemical predictions t m second order O(M 2 ) updates of the total energy and pressure, and for the nal momentum changes. The here presented version is rst order in time for simplicity of exposition: i h n+1 ? (m  + tm ~ 1 p(2);n+1 ? p ~ ) = ?
t r m ~ ~   n+1 (17) p m ~ (2) ;  2 ? e + te = ?
t r~ 2
[H m~ n+1] ; M ?1 + 2 (2)

where

2

H  = 1 (e + p) + 2 t e : 



(18)

Here a single -superscript denotes data after the rst split step, i.e., after the solution of System I , while a double star  indicates data additionaly modi ed by the predictor step. The symbols r~ 1; r~ 2
, denote suitable discrete approximations of gradient and

divergence. The term
tr~ 1 p makes up for the O(M 2 ) dierence between System I and System I , as discussed above. Upon elimination of m ~ n+1 from (17) a discrete Poisson-type equation is obtained for p(2) with a weak O(M 2 ) nonlinearity due to the time dependence of the kinetic energy. This can be accounted for, e.g., by an outer iteration as described in [8]. The Poisson-type equation itself is solved using a classical CG-method from the LINSOL-package [17].

3.7 Compressible projection method The above presented approach can be understood as a compressible projection method or fractional step scheme. Incompressible projection methods have rst been introduced by Chorin [2],[3], while the fractional step scheme approach was introduced by Temam [18],[19]. In the rst step, density and velocity are transported by solving two advection equations without strictly enforcing the divergence constraint of incompressible

ows ~v = ~vn ?
t [(~v
r)~v] (19)  = n ?
t [~v
r] :

The superscript n denotes values at the old time level. In the second step, the intermediate velocity eld ~v is then projected onto the space of divergence-free vector elds. The Hodge-Helmholtz decomposition [15] views an arbitrary vector eld as being composed of two orthogonal components, one divergence-free and the other the curl-free gradient of a scalar eld. The vector eld ~v , de ned on a spatial domain , can thus be written as

~v = ~vd + r

(20)

where r
~vd = 0, and ~vd satis es the boundary conditions.  is given as the solution to the elliptic equation L  = r
( 1 r) = r
~v ; (21) @ = ~v 
nj : @

@n @

The velocity at the new time level tn+1 is then obtained from ~vd;n+1 = ~v ? r. In the above framework the scalar  can be identi ed with the second order pressure term p(2). In the zero Mach number limit and under the assumption, that there is neither a global compression nor a chemical reaction, the predictions vanish (t m ~ = t e = 0) and (17) yields (see also [10])

r~ 2

t r~ 1p(2);n+1 = r~2
~v !

with

~v = ~v +
t r~ 1 p:

(22)

This is the pressure equation of a projection method for incompressible ows [6]. Consider now a projection method that is extended to the compressible equations without the introduction of multiple pressure variables. In the limit M ! 0 the Poisson-type equation will take the form [10] !

r~ 2
1 r~ 1pn+1 = 0:

(23)

Hence the O(M 2 ) pressure uctuations that are known to occur have to be represented numerically by subtle cancellations of large numbers and the performance of such a scheme will depend in an undesired way on the machine accuracy for small Mach numbers. In order to guarantee the proper limit behavior M ! 0 of the incompressible equations, it is necessary to decompose and thereby separate the pressure p into a second order pressure p(2) and the actual thermodynamic pressure P0 within the numerical framework.

3.8 Extension of incompressible methods Besides compressible ow solvers also incompressible solvers can be extended to the regime of low Mach number ows. Numerical schemes for incompressible ows are usually based either on projection methods (e.g., Chorin [2],[3]) or on segregated methods (e.g., the SIMPLE algorithm of Patankar and Spalding [20]). The principle of both methods is a decoupling of pressure and velocity at the new time level tn+1 . Since the above presented scheme can be understood as a compressible projection method, it is possible to apply techniques originally developed for incompressible ows to the low Mach number compressible regime. In the following, an outline of a compressible projection method in terms of primitive variables is given. The compressible Euler equations in primitive variables read as

t + r
(v) = 0 1 vt + (v
r) v + M  rp = 0 pt + v
rp + pr
v = 0 2

:

(24)

The fractional step scheme for (24) separates the convective terms, which are explicitly discretized, t + r
(v) = 0 vt + (v
r) v = 0 ; (25) pt + v
rp = 0 from the sonic terms vt + M1  rp = 0 : (26) p + pr
v = 0 2

t

The following considerations are focused on the second step (26) and rst order accuracy in time. A fully implicit approximation to (26) with pressure decomposition has the form v

n+1

?v


t

n

P0 +1 ?P0
t n

n

+ + MP

n+1  1 (1) 2 (2) ~ P + MP + M p r 2 +1 1 0

M  (1);n+1

n

?P


t

(1);n

+ M 2 p(2)

;n+1

?p(2)


t

;n

= 0

+ pn+1r~ 2
vn+1 = 0

: (27)

The variables after the convection step are represented by vn and pn, as long as no mistaking for the values at the old time level can occur. Because the density remains unchanged in this step, n+1  n. A common procedure to solve system (27) is to separate vn+1 in equation (27)1 and insert the result into (27)2. This gives a nonlinear elliptic equation for the pressure, (due to the coecient p of r~ 2
vn+1) which needs to be linearized. This pressure-correction algorithm can be formulated as:

1. Decompose the pressure pn into pn = P0n + MP (1);n + M 2 p(2);n by suitable averaging procedures. 2. Estimate global wave length eects during the time step to obtain P0; P (1); v. 3. Estimate the pressures p and p(2); , at the next time level, e.g., p = pn +
tP0 +
tMP (1) and p(2); = p(2);n. 4. Calculate the corresponding velocity v from (27)1 by use of rxP (1) =M = r P (1) =  vt as n+1 v = vn +
t n+1 vt ? 
n+1t r~ 1p(2); : (28) 5. Introduce the corrections p(2);n+1 = p(2); + p(2) and vn+1 = v + v into (27), eliminate v, and solve the obtained elliptic pressure-correction equation for p(2) : 2

pr~ 2
 1 r~ 1 p(2) ?
Mt p(2) = 1 (1) + p r ~ 
t P0 + MP
t 2
v : 6. Add the pressure-corrections 







n+1

(29)





2

p(2);n+1 = p(2); + p(2)

and

pn+1 = p + M 2 p(2);n+1 ? p(2);n ; 



(30)

and calculate the corresponding velocity vn+1. The factor pn+1 in (27)2 has been replaced by p. Therefore, steps 4. to 6. should be applied iteratively until convergence is reached. A relaxation parameter  2 [0; 1] can be introduced in step 6. to set p(2);n+1 = p(2); + p(2).

4. NUMERICAL SIMULATIONS 4.1 Inviscid vortex transport at M=0 For zero Mach number no acoustic pressure waves can be accomodated in the domain and the ~-scale variable becomes void. The numerical method then reduces to an incompressible projection method. Gresho and Chan [22] proposed the inviscid transport of a vortex as test case for incompressible constant density ows. A vortex is transported in x-direction through a rectangular domain. The exact solution to such a problem is known, i.e., a pure translation at unit speed of the initial conditions. The boundary conditions for the test case are inlet to the left, outlet to the right, and solid frictionless walls at the top and bottom. The initial conditions are that of a triangle vortex. The tangential velocity component describes a solid body rotation in the core of the vortex. At r = R the velocity then switches to a decreasing linear function of r until r = 2R, where the tangential velocity returns to zero. The vortex initial conditions, before adding the translational

velocity, read as

u (r) = u0r=R u (r) = u0(2 ? r=R) u = 0

0  r  R; R < r  2R; r > 2R;

for

(31)

where u0 = 1. A uniform mesh of 80x20 elements is used to cover the domain that is one unit high and four units long. The radius of the triangle vortex is resolved by 2R = 8
x. In compliance with [22], a CFL number of 0:1 was chosen. Figure 1 shows the relative streamlines of the vortex as it moves through the mesh. The vortex remains centered along the x-axis, and no phase lag can be observed. The in uence of the outlet region deforms the vortex, which is due to re ections from the outlet. The results presented by Gresho and Chan, [22], show a falling vortex, while their SCM-projection method slightly better preserves the maximum of the velocity.

y

1.0

0.5

0.0

Fig.1: Relative streamlines at t = 0, 1, 2, 3; = 0.02 (0.02) 0.18; Mach number M=0 0.0 with MPV-Approach. 1.0 2.0 3.0 x

The test case clearly demonstrates the zero Mach number capability of the MPVApproach. Improved outlet boundry conditions for the pressure p(2) are being developed to obtain low re ection boundaries.

4.2 Generation of baroclinic vorticity at M=1/20 For nonzero Mach numbers, the acoustic scale ~ is coupled with the small scale ~x by the linearized acoustic source term in the momentum equation (7)2. An acoustic pulse with amplitude P (1) will lead to the creation of baroclinic vorticity due to compressibility eects.

4.0

In a double-periodic domain of size 40x8 units, the following initial data are given:

(x; y; 0) p(x; y; 0) u(x; y; 0) v(x; y; 0)

1 = 0 + M ~(1) 0 2 (1:0 + cos(x=L)) +(y ) 1 = p0 + M p~(1) 0 2 (1:0 + cos(x=L)) = u~0 21 (1:0 + cos(x=L)) = 0:0

for

?L  x  L = M1 1 0  y  Ly = (2L)=5 ; M = 20 ;

and the coecients

0 = 1:0 (0) ~0 = ~(1) 0 = 0:4

p0 = 1:0 p~(1) 0 = 2

u~0 = 2p :

9 > > > > = > > > > ;

(32)

(33)

The function (y) in (32) is de ned by

y ; 0  y  1L (y) = y (y ? 1 L ) ? ~(0) ; 1 L < y  2L y ; with y = 2~(0) 0 =Ly : (34) y y 0 2 y 2 y (

The data from (32) represent a periodic train of long-wave right- running acoustic pulses, that set a saw-tooth like density layering into motion. The numerical solution uses 401 x 80 grid points within the interval of (32)2 , which gives a constant background pressure P0. The acoustic equations (9) are solved using a Crank-Nicholson type approximation, while the coupling term has been neglected. The dierent density of two neighboring particles leads to a dierent speed under the in uence of the same acoustic pressure wave. This results in the occurence of a Kelvin-Helmholtz instability [16] where one uid is moving at a dierent rate relative to another. Figure 2 shows, that the initially horizontal interface starts rolling up into large vortical structures, creating vorticity. The large scale acoustics thereby feed energy into the small scale structures. The computations were performed on a CRAY J90.

Fig.2: Density isocontours at values of 0.8 and 1.2 on a 401x80 grid. Mach number M=1/20 and times t = 7.0, 15.0 with MPV-Approach.

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