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Artificial dissipation methods for Euler equations V. Selmin

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## Artificial dissipation methods for Euler equations V. Selmin

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Artificial dissipation methods

for Euler equations

V. Selmin

Multidisciplinary Computation and Numerical Simulation

Conservation law form:

Unknowns & Flux vector:

State equation:

Non conservative form:

Euler equations

Hyperbolicity:

Let be a linear combination of the fluxes where is a nonzero vector.

The system is hyperbolic if the the Jacobian matrix has only real eigenvalues.

Eigenvector and eigenvalues:

The eigenvectors form the matrix that diagonalises :

Jacobian matrix decomposition:

The following decompositions is also introduced:

The matrices and are diagonalised by the transformation :

Flux homogeneity

Euler fluxes are homogeneous of degree one with respect to W

Euler equations

Node-Centred finite volume

Discretisation of the integral equation:

Discrete equation:

Consistency:

Conservative systems:

Second-order spatial approximation:

Finite Volume/Finite Element Formulation

Positivity

Scalar equation:

Discrete equation:

Condition on scheme positivity:

Monotone numerical scheme:

Extension to systems of equations:

Finite Volume/Finite Element Formulation

The weak solutions are not univocally defined for the partial differential problem: different solutions may satisfy the initial condition.

An additional principle is then needed for the settlement of a solution physically acceptable.

Let consider the hyperbolic equation

where f(u) is a nonlinear function of the uknown.

The eligible solution is the limit when of the solution of the viscous problem:

Oleinik has shown that the eligible solution must satisfy the following inequalities

for . These relations are called entropy condition.

Entropy condition

A central role is played in the algorithm by the modelling of the interactions of fluxes at the interfaces of adjacent

grid cells.

Two distinct sets of state quantities must be combined into one single set of numerical fluxes at the interface.

In the following, some upwind related tecnologies which may be used to build the numerical flux will be discussed.

Flux Difference Splitting:

Let consider the Riemann problem governed by the one-dimensional inviscid flow equation:

Roe’s scheme may be interpreted as an approximate Riemann solver based on the linearised equation:

The matrix has to satisfies the following properties:

1- (Consistency)

2- has a complete set of real eigenvalues and eigenvectors (Hyperbolicity)

3- (Conservation propertty)

4- When the two states are connected through one discontinuity, the discontinuity can be reproduced exactly.

Conditions 1 and 3 have to be satisfied when the Jacobian is computed at an average state:

Flux Difference & Vector splittings

Roe introduced the following average operator for a polytropic ideal gas:

The average operator is in general valid for a gas whose pressure law allows the flux F to be an homogeneous

function of degree one in the conservative variables. As the pressure is expressed in the general form

that means the following condition has to be satisfied:

The numerical flux on this approximate Riemann solver may be expressed by

Another approximate Riemann solver , with similar properties, is due to Osher:

Numerical flux not differentiable

Crisp representation of discontinuities

Does not satisfy the entropy condition

Allows expansion shocks

Numerical flux differentiable

Integration path // to the eigenvectors of A

Satisfy the entropy condition

Flux Difference & Vector splittings

In the case of Flux Vector Splitting (FVS), the numerical flux across an interface separating two states and

is considered as the sum of two parts and that should possess the following desirable properties:

1- (consistency to the governing equations),

2- must have all the eigenvalues , must have all the eigenvalues

3- must be continuous with for , for where M is the

Mach number normal to the interface

4- The components of and together must reproduce the symmetry of F with respect to M (all other

quantities held constant), that is

if

5- must be continuous

Restriction (3) ensures that in supersonic regions leads to standard upwind differencing.

In general, the flux decomposition is chosen in such a way that are homogeneous of degree one.

The numerical flux may be expressed as

which can be rewritten to make it to look like the Osher’s formula

The integration is independent of the path since is a perfect gradient

Flux Difference & Vector splittings

Mach number contours

Symmetry axis+body nodes

Computational domain (65x41) nodes

Test case: hypersonic flow around a cylinder

The test case corresponds to compute a flow around a cylinder section.

In this case, a detached choc of high intensity (from M=8 to M=0.3929) is present in front of the body.

The flow, being subsonic behind the shock, becomes supersonic at approximatively 29% of the cylinder radius.

Flux Difference & Vector splittings

“Carbuncle” effect in Roe scheme

Roe’s scheme & the entropy condition

As already anticipated previously, the numerical flux of the Roe scheme may be written for multidimensional

problems according to

where the quantities in the dissipation term are computed by using the average proposed by Roe and

To have a zero value of the eigenvalues independently of the difference may lead to a violation of

entropy condition, whereas the dissipation matrix may become singular when .

Flux Difference & Vector splittings

not satisfied

Entropy condition

satisfied

Roe’s scheme & the entropy condition

Harten proposed to regularize the dissipation matrix by using the following correction of the eigenvalues:

Flux Difference & Vector splittings

A method to extend the upwind schemes to multidimensional problems is based on the property of the

Euler’sequations to be invariant with respect to a rotation of the coordinate axis.

Let be a linear combination of the fluxes where is a nonzero vector of real quantities.

A local rotation of the velocity vector is defined in the following way:

The state is transformed into the state

Moreover, applying the the rotation to the flux allows to describe the flux vector in a one dimensional fashion

Finally, the inverse transformation express the different flux in the original frame

The eigenvalues of the Jacobian matrix are not modified

Flux Difference & Vector splittings

The matrix dissipation flux

may be described in a straightforward way for the Euler equations.

In the rotated frame, the matrix dissipation operator takes the form

where

It may be demonstrated that by using Roe’s averages

and , we obtain

Matrix Dissipation Operator

The matrix dissipation flux becomes

The coefficients are functions of the eigenvalues of the system and take the form:

which may be expressed with respect to the Mach number normal to the interface according to

Matrix Dissipation Operator

- The first term of the matrix dissipation operator is able alone to maintain the monotonicityof Roe scheme
- The second term is a diffusive operator acting only in subsonic regime.
- The third term is an antidiffusive operator that acts both in subsonic and supersonic regime.

Matrix Dissipation Operator

Matrix Dissipation Operator

In order to enforce the entropy condition, the coefficients may be regularised according to

Matrix Dissipation Operator

A desirable property of a numerical scheme for the solution of the Euler equations is its capability to reproduce an

isoenergenetic steady state flow.

Whether the discretisation of the convective flux might provide a solution which satisfies the solution, this no more

true when the dissipation operator is added to the numerical scheme.

To obtain a scheme which satisfies this property, the following dissipative flux is introduced

Properties:

1- The flux preserves the monoticity of the numerical flux.

2- Essential property in order to obtain accurate stagnation temperature

3- Enthalpy damping techniques may be used to speed up convergence of transonic flows solution.

Total enthalpy constant over the entire field

Matrix Dissipation Operator

A more compact expression of may be obtained by rewritting the scheme with respect to the coefficients

and only.

Characteristic variables:

may be expressed according to

Matrix Dissipation Operator

In the context of centered approximations, a simpler approximation for the dissipation operator may be

obtained by replacing in the matrix dissipation operator all the eigenvalues of the Jacobian by its spectral radius:

The modification leads to the following scalar dissipation operator

which preserves the monoticity of the scheme,but it is more dissipative. In the original frame, the scalar dissipation

operator may be transformed according to:

Scalar Dissipation Operator

Van Leer Flux Vector Splitting

A very promising study for splitting the flux has been introduced by Van Leer. He deduced formulas that satisfy the

basic requirements , but with the following additional properties:

1- must have one eigenvalue vanishing for .

2- , like , must be polynomials in M and of the lowest degree.

Constraint (1), that van Leer considered crucial, makes it possible to build numerical shock structures using two

interior zones. The requirement (2) makes the splitting unique.

The formulas for are given in terms of the Mach number.

For supersonic and sonic flows, , we have

and for subsonic flow, ,

Split fluxes continuously differentiable

Valid only for a perfect gas

Flux Vector Splitting

Montagné Flux Vector Splitting

A more physical way to build a flux splitting consists in approaching the conservative laws by a collisionless

Boltzmann equation. Each state is split into group of particles which are distributed according to their velocities.

The density law of particles, which obeys a Maxwell distribution in the kinetic theory of gases, must be approximated

by a simpler distribution law. The fluxes and are defined by considering the particles with positive and

negative velocities, respectively. In addition, it is requested that reduces to F for supersonic flows with a

positive (negative) velocity.

The formulas for are given in terms of the Mach number.

For supersonic and sonic flows, , we have

and for subsonic flow, ,

Split fluxes not always continuously differentiable

Valid not only for a perfect gas

Property (1) not satisfied

Flux Vector Splitting

Enthalpy preserving Flux Vector Splitting

The new splitting is based on the following heuristic consideration:

1- The fluxes of van Leer scheme are continuously differentiable and are expressed in a general form (OK).

2- The fluxes of the Boltzmann type scheme are constinuously differentiable and are very close to the

corresponding fluxes of the van Leer type (OK).

3- For the components of the third fluxes, any of the the previous schemes can be retained. In addition, they do

not satisfy the condition of constant total enthalpy for isoenergenetic steady flows. They can be repaired by

taking . Moreover, if fluxes are continuously differentiable, it will also the case for .

For supersonic and sonic flows, , we have

and for subsonic flow, ,

Split fluxes continuously differentiable

Valid for any gas

Very simple expressions

Flux Vector Splitting

Applying the rotation to the flux allows to split the flux in a one-dimensional fashion:

After splitting , the inverse transformation

gives the following relations for the flux splitting:

Then the numerical flux reads

For multidimensional applications, the proposed flux splitting is described by the following formulas.

Indicating by the quantity , for , we have

and for subsonic flow,

Flux Vector Splitting

j

i+2

i

i+1

i-

i

i-1

Extension to second order spatial accuracy

All upwind schemes (FDS, FVS) may be interpreted as built from a centred discretisation (second-order accurate)

spatial approximation + an artificial dissipation term (first-order accuracy)

The schemes are first-order accurate in space.

In general, artificial dissipation is needed only at discontinuities, thus development of sensors that identify

discontinuities (or rapid change of the solution) and monitor the level of dissipation by using the sensor.

The quantities and are replaced by the following expressions

where s and are parameters,

and are “extrapolated” values of the unknowns.

N.B.: The values of and may be used in the evaluation of all the terms of the FDS or FVS schemes or only for the evaluation of the dissipative term.

FDS

FVS

Second order accuracy

Extension to second order spatial accuracy

The variation may be expressed according to

Thus,

Then, in order to avoid oscillations in the numerical solution, in particular near shocks, the extrapolation has to be

limited by varying the switching factor s , which is controlled by the solution itself.

Typical formulation of such limiters are given by

where is a small number ( ) preventing division by zero in regions of null gradients.

These limiter formulations react to the local gradient and to the curvature of the solution, i.e.

Thus, in regions of weak changes of W , the limiter remains nearly to one, but for strong changes the value

decreases and the scheme reduces the accuracy generating a stronger numerical simulation.

First-order scheme

Higher-order scheme

Second order accuracy

j

ej

i

ei

i-

j*

i*

Extension to second order spatial accuracy

Methods to computed extrapolated variables

1- Centred gradient:

This formulation works well for subsonic and transonic flows, but not for high

supersonic flows, for which it is not able to forecast the correct slope of the

solution upstream or downstream of the nodes i and j, respectively, since the

gradient approximation is centered.

2- Upstream/downstream element gradient:

This formulation works well for subsonic, transonic flows and supersonic flows.

It forecasts the correct slope of the solution upstream or downstream of the node i and j, respectively.

Nevertheless, it needs additional data structures for the computation of the gradient on the element.

3- Upstream/downstream segment variation:

This formulation works well for subsonic, transonic flows and supersonic flows.

It forecasts an approximation of slope of the solution upstream or downstream of the node i and j, respectively.

It gives a good information for the computation of the limiter s, but the approximation for the third-order operator is

less accurate. It needs very few additional data structure information to work.

The nodes i* and j* are selected in such a way that

the segments i-i* and j-j* minimise the scalar product

with the segment i-j

Second order accuracy

Extension to second order spatial accuracy

- Alternative methods to compute the third-order term
- where
- Remarks:
- 1- The use of the operator leads to a final discretisation similar to those of a directional fourth-order derivate
- 2- The use of the operator leads to a final discretisation similar to those of a biharmonic operator
- 3- The use of the harmonic operator leads to a more robust discretisation for multidimensional grids with deformed
- elements.

Second order accuracy

Extension to second order spatial accuracy

Special treatments for the scalar dissipation

In the classical scalar artificial method, it is preferred to use the following expression for the variation:

with

where the discontinuity sensors are expressed in terms of pressure variations

Remarks:

1- The aim of the background dissipation is to improve the ability of the scheme to damp high frequency modes in

smooth part of the flow.

2- are adjustable parameters.

3- The value of is limited to 1 because we know that with this value the scheme maintains monoticity.

4- The term is substract from to cut off the fourth order difference operator, which could otherwise

cause an oscillation in the neighbourhood of a shock wave.

shock capturing

background dissipation

Second order accuracy

At hypersonic speed, a strong shock appears in front of an aerospace vehicle.

The compression of the gas through the shock is associated with an increase in temperature.

Through the shock wave , the total enthalpy is constant and along a streamline, we can write

If the velocity is very large, .

A strong shock causes a large variation of the fluid velocity. Thus, in front of the vehicle, .

Consequently, we have

This means that almost all the kinetic energy is transformed into heat, resulting in a very high temperature in the

region between the body and the shock. An important characteristic of hypersonic phenomena is then the

transformation of an hypervelocity flow into an hyperenthalpy one.

Another characteristic is that a part of that energy is transformed into chemical energy, as a consequence of

(i) excitation of the internal degree of freedom of molecules,

(ii) dissociation of molecules,

(iii) ionisation.

These phenomena are endothermic and the final temperatures reached are much lower than those obtained for

a non-reactive gas. Moreover, in these conditions, the properties of air are modified.

The chemical composition of air can be approximately subdivided into the following regimes:

1- T<2500K. The chemical composition is substantially that at room temperature.

2- 2500<T<4000K. The oxygen dissociation regime, no significant nitrogen dissociation,slight nitric oxyde formation

3- 4000<T<8000K. The nitrogen dissociation regime, oxygen fully dissociated.

4- t>8000K. Inonization of the atomic constituents.

h: specific enthalpy

Modelling of high speed flows

= energy of formation

= mass fraction

= specific heat at constant volume

= energy of vibration

= universal gas constant

= molar mass

Modelling of High Speed flows

The model of a real gas can be described as a mixture of perfect gases which are subject to chemical reactions.

The state law must take into account , on one hand, the species decomposition and, in another hand, the

exitation of internal degrees of freedom of atoms and molecules (translation, rotation, vibration).

The internal energy takes the form

Note that

Using Dalton’s law, the pressure is given by

The specific enthalpy is written according to

Modelling of high speed flows

The following alternatives for the definition of the ratio of specific heats may be introduced:

1- True ratio of specific heats

2- Equivalent relating pressure to internal energy and density

3- Equivalent relating sound speed to pressure and density

4- Ratio of specific heats of the frozen flow

Modelling of high speed flows

A gas formed by standard air (oxygen+nitrogen) is assumed and the vibrational degrees of freedom of the diatomic

molecules are neglected. We obtain a perfect gas with constant ratio of specific heats, i.e.

The pressure and the sound speed can be expressed in a simple way as a function of the conservative variables:

Remarks:

1- The upwind schemes described previously have been build in such a way they satify the condition of total

enthalpy at steady state. That is true for first-order scheme. But, special care must be taken when employing

the second order one. The standard procedure is to adopt the primitive variables as sensor for

the limiters, which does not satisfy the property. The total enthapy per unit volume has to be used

instead of pressure.

2- The upwind formulas depends of five quantities . The limiting procedure is performed on

only three of them. This is possible because a constant specific heats ratio is assumed.

3- For hypersonic flows, a drawback is the use of the static speed of sound that control the eigenvalues.

In strong expansions, this quantity becomes very small. In this case, it is suggested to use the critical sound

speed instead:

which does not influence the total flux F and approaches the static speed of sound at the sonic points.

4- The critical sound speed is proportional to the square root of the total enthlapy and is therefore constant

for isoenergenetic Euler flows. That leads to a gain in robustness of the solver

Modelling of high speed flows

The laws governing chemical reactions have been coupled with the Euler solver. At each iteration of the time

integrator, the Euler code provides and update for . The mass fractions are computed by solving a

set of nonlinear algebaric equations in the case of chemical equilibrium and a set of partial differencial equations

in the case of chemical non-equilibrium. The temperature is computed by expressing the identity of the internal

specific energy written in terms of “fluid motion quantities” and “chemical ones”:

The outcome is used to update the other thermodynamic quantities in preparation of the next step.

The desirable properties that a FVS schemes should possess are:

1- (consistency to the governing equations)

2- must be continuous, with

3- must have all eigenvalues

must have all eigenvalues

Remarks

1- In the definition of the upwind we have a switch at . The efficiency of the scheme depends on the

definition given to the sound speed.

Property ( 1) is always satisfied independently of the definition of the sound speed.

To ensure that in supersonic regions property (2) is satisfied, we might use the sound speed as it appears in

the true eigenvalues of the system:

Modelling of high speed flows

2- Some authors prefer to employ a “false sound speed” approach based solely on satisfaction of property (1):

whose value is closer to that obtained from available experimental data.

3- Other choices for the definition of the sound speed are a priori possible. However property (3) has to be satisfied.

The consequence is that and can not vary independently. In particular, it can be shown that must be

greater or equal to .

4- Contrary to non-reactive flows, the specific heats ratios for reactive flows are no longer constant.

Then, it is propose to perform the limiting procedure also to and for second-order accurate computations.

5- The concept of critical sound speed has been extended to reactive flows according to

Modelling of high speed flows

Computational domain (65x41) nodes

Iso-Pt contours

Non reactive flows around a blunt body

A first test of the numerical scheme has been made for a flow around a half cylinder at .

The freestream conditions are taken to approximatively correspond to the standard atmosphere

at the altitude of 75 km ( ).

1- scheme a: first-order scheme

2- scheme b: second-order scheme with limitation on

3- scheme c: second-order scheme with limitation on

Modelling of high speed flows

Entropy generation along stagnation streamline

Total enthalpy residual versus iterations

Stagnation temperature versus iterations

Non reactive flows around a blunt body

Better accuracy of the second

order schemes (b & c).

Sheme c allows better accordance

with respect to entropy generation

and stagnation temperatures

levels

Modelling of high speed flows

flow

Reactive

flow

Iso-Cp contours

Non reactive

flow

Reactive

flow

Iso-Temperature contours

Reactive flows around a blunt body

The results obtained on the previous case are illustrated now with the hypothesis of reactive flow in thermodynamic

equilibrium. Comparisons with non-reactive flow results show that the shock has considerably moved towards the

body. The temperature level is considerably reduced and the stagnation temperature is now approximatively 22%

of the corresponding non-reactive flow stagnation temperature.

Diatomic

Oxygen

Diatomic

Nitrogen

Mass fractions for Oxygen and Nitrogen

Modelling of high speed flows

1- Augmented dissipation schemes have been developed for generalised finite volume schemes.

2- Flux Difference Splitting and Flux Vector Splitting schemes may be re-interpreted as

a central scheme with augmented dissipation operator, but are first-order accurate schemes

3- Augmented dissipation has to be activated where discontinuities are present, but has to vanish in regions

where the solution is regular

4- Numerical schemes have been developed in such a way that total enthalpy is constant for iso-energenetic

steady flows (needed in order to obtain accurate stagnation temperature value)

In particular, the variables on which limitation is performed have to be accurately chosen.

5- The treatment of high enthalpy non-reactive and reactive flows has been discussed.

→extension to finite elements schemes is automatic

→introduction of limiters and shock sensors

Modelling of high speed flows

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