**Space-time discontinuous Galerkin method for the numerical simulation of**

**viscous compressible gas flow with the k-omega turbulence model**

*Jan* ˇCesenek1,a

1_{VZLU - Czech Aerospace Research Center, Beranov´ych 130, 199 05 Praha - Letˇnany, Czech Republic}

**Abstract.** In this article we deal with the numerical simulation of the non-stationary compressible turbulent
flow described by the Reynolds-Averaged Navier-Stokes (RANS) equations. This RANS system is equipped with
two-equation k-omega turbulence model. The discretization of these two systems is carried out separately by the
space-time discontinuous Galerkin method. This method is based on the piecewise polynomial discontinuous
approximation of the sought solution in space and in time. We use the numerical experiments to demonstrate the
applicability of the shown approach. All presented results were computed with the own-developed code.

**1 Introduction**

During the last decade the space-time discontinuous
Ga-lerkin finite element method (ST-DG), which is based on
piecewise polynomial discontinuous approximations of the
sought solution, became very popular in the field of
numer-ical simulation of the fluid flow. This method of higher
or-der was successfully used for the simulation of the
Navier-Stokes equations [1–6]. In the case of compressible
tur-bulent flow the finite volume - space-time discontinuous
Galerkin method was used [7–9], where the equations of
the turbulence model were discretized by the implicit finite
volume method. This article is devoted to the
discretiza-tion of viscous compressible turbulent gas flow using
ST-DG applied also for the equations of the turbulence model.
The flow is described by the system of the RANS
equa-tions equipped with the system of*k*−ωequations. These
two systems are solved separately and discretized by
ST-DG. In this article Wilcox*k*−ωturbulence model was
cho-sen [10]. The flow around airfoil RAE2822 will be used
for the numerical simulations. These results are compared
with experiments. Computational results show that the
ST-DG method is good approach for the solution of these types
of problems.

**2 Formulation of the**

*k*

_{−}

### ω

**turbulence**

**model**

We consider compressible turbulent flow in a bounded
do-mainΩ_{⊂}*IR*2_{. We assume that the boundary}_{Ω}_{consists of}

three disjoint parts∂Ω = ΓI _{∪}ΓO_{∪}ΓW, whereΓI is the
inlet,ΓOis the outlet andΓWis impermeable wall.

The system of the RANS equations describing the vis-cous compressible turbulent flow can be written in the form

∂w
∂*t* +

2

*s*=1

∂f*s(*w)
∂*xs* +

2

*s*=1

∂f*sp*(w)
∂*xs*

a _{e-mail:}_{[email protected], [email protected]}

=

2

*s*=1

∂k*s(*w)
∂*xs* +

2

*s*=1

∂R*s(*w,∇w)

∂*xs* , (1)

where for*s*=1,2 we have

w=(w_{1}, ..., w4)*T* =(ρ, ρv1, ρv2,*E*)*T* ∈*IR*4,

f*s(*w)=(*fs*,1, ...,*fs*,4)*T*

=(ρvs, ρv_{1}vs+δ1*sp*,
ρv2vs+δ2*sp*,(*E*+*p*)vs)*T*,

f*sp*(w)=(*f _{s}p*,1, ...,

*f*

*p*
*s*,4)*T*

=(0,2

3ρ*k*δ1*s,*
2
3ρ*k*δ2*s,*

2
3ρ*k*vs)*T*,

R*s(*w,∇w)=(*Rs*,1, ...,*Rs*,4)*T*

=

0, τ*Vs*1, τ*Vs*2,
2

*r*=1

τ*V _{sr}*vr+

_{c}_{pµL}

*Pr* +
*cpµT*

*PrT*

∂θ
∂*xs*

*T*

,

k*s(*w)=(*ks*,1, ...,*ks*,4)*T*

=

0, ...,0,

µL+σkρ*k*

*e*ω˜

∂*k*

∂*xs*

*T*
,

where

τ*Vsr*=−2_{3}(µL+µT) divuδsr+2(µL+µT)*dsr(*u),

*dsr*(u)= 1
2

∂vs
∂*xr* +

∂vr
∂*xs*

.

We use the following notation: u = (v_{1}, v2) - velocity, ρ

- density, *p*- pressure,θ- absolute temperature, *E*- total
energy,γ- Poisson adiabatic constant,κ- heat conduction
coefficient,*c*v- specific heat at constant volume,*cp*-
spe-cific heat at constant pressure, where*cp* = γ*c*v,*Pr*is the

laminar Prandtl number, which can be express in the form

formula. The above system is completed by the thermody-namical relations

*p*=(γ−1)

*E*−1_{2}ρ|u_{|}2_{−}ρ*k*

, θ= 1

*c*v

_{E}

ρ −
1
2|u|2−*k*

and is equipped with the initial condition

w(*x*,0)=w0(*x*), *x*∈Ω
and the following boundary conditions

a) ρ|ΓI =ρD,

b) u_{|Γ}* _{I}* =u

*D*=(vD1, vD2)

*T*,

c)

2

*s*,*r*=1

τ*V _{sr}nsvr*+

_{c}_{pµL}

*Pr* +
*cpµT*

*PrT*

∂θ

∂*n* =0 onΓI,

a) u_{|ΓW} =0,

b) ∂θ
∂*n*

_{Γ}* _{W}* =0,

a)

2

*s*=1

τ*V _{sr}ns*=0,

*r*=1,2 onΓO, b) ∂θ ∂

*n*

_{Γ}* _{O}* =0,

with given dataw0_{, ρD,}_{u}_{D. It is possible to show that}_{f}_{s(α}_{w}_{)}_{=}

αf*s(*w) forα >0. This property implies that

f*s(*w)=A*s(*w)w, *s*=1,2, (2)

where

A*s(*w)= *D*f*s(*w)

*D*w , *s*=1,2,

are the Jacobi matrices of the mappingsf*s. Similary we can*
express

f*sp*(w)=A*sp*(w)w, *s*=1,2.

The viscous termsR*s(*w,∇w) can be expressed in the form

R*s(*w,_{∇}w)=

2

*k*=1

K*s*,*k(*w)

∂w

∂*xk*, *s*=1,2, (3)

whereK*s*,*k(*w)∈*IR*4×4are matrices depending onw.

The aforesaid system of the RANS equations is com-pleted by the Wilcox’s turbulence model (see [10]) forµT. For the sake of the stability we introduce new variable

˜

ω=*ln*ω. (For details see [11]). Then the turbulence model

reads

∂ρw˜
∂*t* +

2

*s*=1

∂˜f*s( ˜*w)
∂*xs* =

2

*s*=1

∂R˜*s( ˜*w,∇w˜)

∂*xs* +s˜( ˜w), (4)

where for*s*=1,2 we have

˜

w=( ˜w1,w˜2)*T* =( ˜ω,*k*)*T* ∈*IR*2,

˜

f*s( ˜*w)=( ˜*fs*,1,*f*˜*s*,2)*T* =(ρωvs, ρ˜ *k*vs)*T*,

˜

R*s( ˜*w,_{∇}w˜)=( ˜*Rs*,1,*R*˜*s*,2)*T* =

(µL+σωµT)

∂ω˜

∂*xs*,(µL+σkµT)
∂*k*

∂*xs*

*T*
,

˜

s( ˜w)=(αω*Pk*

*k* −βρ*e*ω˜ +
2

*s*=1

(µL+σωµT)

∂ω˜
∂*xs*

∂ω˜
∂*xs* +

˜

*CD,*

*Pk*−β∗ρ*e*ω˜*k*)*T*.

Here ω is the turbulence dissipation,*k* is the turbulence
kinetic energy andµT is the eddy viscosity. We can write
the production term as

*Pk*=

2

*s*=1
2

*r*=1

τ*Tsr*∂vs_{∂}_{x}*r*,

where

τ*Tsr*=−2_{3}µTdivuδsr−2_{3}ρ*k*δsr+2µT *dsr(*u),

*dsr(*u)= 1

2

∂vs
∂*xr* +

∂vr
∂*xs*

,

µT = ρ*k*

ωlim.

Limited eddy viscosityωlimis given by the term

ωlim =max

*e*ω˜,*Clim*

1 2β∗

2

*r*,*s*=1

˜ τrsτrs˜

, ˜

τrs =−2

3divuδrs+2*drs(*u).

The cross-diffusion term*CD*is defined as

*CD*=σD* _{e}*ρ

_{ω}

_{˜}max

2

*s*=1

∂*k*
∂*xs*

∂ω˜
∂*xs*,0

.

The coefficientsβ, β∗, σk, σω, σD, αω,*Clim,PrT* are chosen
by [10]:

αω= 13

25, β=0.0708, β∗=0.09, σk=0.6, σω=0.5,

*Clim*= 7

8, σD= 1

8, *PrT* =
8
9.

This system is also equipped with the initial condition

˜

w(*x*,0)=w˜0(*x*), *x*_{∈}Ω
and the following boundary conditions

a) ˜ω|Γ*I* =ωD,˜ b)*k*|Γ*I* =*kD,*

a) ˜ω_{|}Γ*W* =ω˜w*all,* b)*k*|Γ*W* =0,

a) ∂ω˜
∂*n*

_{Γ}* _{O}* =0, b)

∂*k*

∂*n*

_{Γ}* _{O}* =0,

with given data ˜w0_{,}_{ωD,}_{˜} _{k}_{D,}_{ω}_{˜}

w*all. Similary like in the RANS*

case we can express convect terms

˜

f*sp*(w)=A˜*sp*( ˜w) ˜w, *s*=1,2 (5)

and diffusion terms

˜

R*s( ˜*w,_{∇}w˜)=K˜*s( ˜*w)∂w˜

∂*xs*, *s*=1,2, (6)

formula. The above system is completed by the thermody-namical relations

*p*=(γ−1)

*E*−1_{2}ρ|u_{|}2_{−}ρ*k*

, θ= 1

*c*v

_{E}

ρ −
1
2|u|2−*k*

and is equipped with the initial condition

w(*x*,0)=w0(*x*), *x*∈Ω
and the following boundary conditions

a) ρ|ΓI =ρD,

b) u_{|Γ}* _{I}* =u

*D*=(vD1, vD2)

*T*,

c)

2

*s*,*r*=1

τ*V _{sr}nsvr*+

_{c}_{pµL}*Pr* +
*cpµT*
*PrT*
∂θ

∂*n* =0 onΓI,

a) u_{|ΓW} =0,

b) ∂θ
∂*n*

_{Γ}* _{W}* =0,

a)

2

*s*=1

τ*V _{sr}ns*=0,

*r*=1,2 onΓO, b) ∂θ ∂

*n*

_{Γ}* _{O}* =0,

with given dataw0_{, ρD,}_{u}_{D. It is possible to show that}_{f}_{s(α}_{w}_{)}_{=}

αf*s(*w) forα >0. This property implies that

f*s(*w)=A*s(*w)w, *s*=1,2, (2)

where

A*s(*w)= *D*f*s(*w)

*D*w , *s*=1,2,

are the Jacobi matrices of the mappingsf*s. Similary we can*
express

f*sp*(w)=A*sp*(w)w, *s*=1,2.

The viscous termsR*s(*w,∇w) can be expressed in the form

R*s(*w,_{∇}w)=

2

*k*=1

K*s*,*k(*w)

∂w

∂*xk*, *s*=1,2, (3)

whereK*s*,*k(*w)∈*IR*4×4are matrices depending onw.

The aforesaid system of the RANS equations is com-pleted by the Wilcox’s turbulence model (see [10]) forµT. For the sake of the stability we introduce new variable

˜

ω=*ln*ω. (For details see [11]). Then the turbulence model

reads

∂ρw˜
∂*t* +

2

*s*=1

∂f˜*s( ˜*w)
∂*xs* =

2

*s*=1

∂R˜*s( ˜*w,∇w˜)

∂*xs* +s˜( ˜w), (4)

where for*s*=1,2 we have

˜

w=( ˜w1,w˜2)*T* =( ˜ω,*k*)*T* ∈*IR*2,

˜

f*s( ˜*w)=( ˜*fs*,1,*f*˜*s*,2)*T* =(ρωvs, ρ˜ *k*vs)*T*,

˜

R*s( ˜*w,_{∇}w˜)=( ˜*Rs*,1,*R*˜*s*,2)*T* =

(µL+σωµT)

∂ω˜

∂*xs*,(µL+σkµT)
∂*k*

∂*xs*

*T*
,

˜

s( ˜w)=(αω*Pk*

*k* −βρ*e*ω˜ +
2

*s*=1

(µL+σωµT)

∂ω˜
∂*xs*

∂ω˜
∂*xs* +

˜

*CD,*

*Pk*−β∗ρ*e*ω˜*k*)*T*.

Here ω is the turbulence dissipation, *k*is the turbulence
kinetic energy andµT is the eddy viscosity. We can write
the production term as

*Pk*=

2

*s*=1
2

*r*=1

τ*Tsr*_{∂}∂vs_{x}*r*,

where

τ*Tsr*=−2_{3}µTdivuδsr−2_{3}ρ*k*δsr+2µT *dsr(*u),

*dsr*(u)= 1

2

∂vs
∂*xr* +

∂vr
∂*xs*

,

µT = ρ*k*

ωlim.

Limited eddy viscosityωlimis given by the term

ωlim =max

*e*ω˜,*Clim*

1 2β∗ 2

*r*,*s*=1

˜ τrsτrs˜

, ˜

τrs=−2

3divuδrs+2*drs(*u).

The cross-diffusion term*CD*is defined as

*CD*=σD* _{e}*ρ

_{ω}

_{˜}max

2
*s*=1
∂*k*
∂*xs*

∂ω˜
∂*xs*,0

.

The coefficientsβ, β∗, σk, σω, σD, αω,*Clim,PrT* are chosen
by [10]:

αω= 13

25, β=0.0708, β∗=0.09, σk=0.6, σω=0.5,

*Clim*= 7

8, σD= 1

8, *PrT* =
8
9.

This system is also equipped with the initial condition

˜

w(*x*,0)=w˜0(*x*), *x*_{∈}Ω
and the following boundary conditions

a) ˜ω|Γ*I* =ωD,˜ b)*k*|Γ*I* =*kD,*

a) ˜ω_{|}Γ*W* =ω˜w*all,* b)*k*|Γ*W* =0,

a) ∂ω˜
∂*n*

_{Γ}* _{O}* =0, b)

∂*k*

∂*n*

_{Γ}* _{O}* =0,

with given data ˜w0_{,}_{ωD,}_{˜} _{k}_{D,}_{ω}_{˜}

w*all. Similary like in the RANS*

case we can express convect terms

˜

f*sp*(w)=A˜*sp*( ˜w) ˜w, *s*=1,2 (5)

and diffusion terms

˜

R*s( ˜*w,_{∇}w˜)=K˜*s( ˜*w)∂w˜

∂*xs*, *s*=1,2, (6)

where ˜K*s(*w)∈*IR*2×2_{are matrices depending on ˜}_{w}_{.}

**3 Discretization**

**3.1 Space discretization of the problem**

ByΩh we denote polygonal approximation of the domain
Ω. LetT*h*be a partition of the domainΩhinto finite
num-ber of closed elements with mutually disjoint interiors such
that Ωh = *K*∈T*hK*. In 2D problems, we usually choose
*K*∈ T*h*as triangles or quadrilaterals. ByF* _{h}*we denote the
system of all faces of all elements

*K*∈ T

*h. Further, we*in-troduce the set of boundary facesF

*B*

*h* ={Γ∈ F*h*;Γ⊂∂Ωh}
and the set of inner faces F*I*

*h* =F*h*\F*hB*. EachΓ∈ F*h*
is associated with a unit normal vector*n*Γ. ForΓ∈ F* _{h}B*the

normal*n*Γhas the same orientation as the outer normal to

∂Ωh. For eachΓ ∈ F*I*

*h* there exist two neighbouring
ele-ments*KL*

Γ,*K*Γ*R* ∈ T*h*such thatΓ⊂∂*K*Γ*L*∩∂*KR*Γ. We use the

convention that*KR*

Γ lies in the direction of*n*Γ and*K*Γ*L*lies

in the opposite direction to*n*Γ. IfΓ∈ F* _{h}B*, then the element

adjacent toΓwill be denoted by*K*_{Γ}*L*.

We shall look for an approximate solution of the prob-lem in the space of piecewise polynomial functions

S* _{h}p*=(

*S*)4,

_{h}p*S*=

_{h}p_{{}v;v

_{|}

*K*∈

*Pp*(

*K*),∀

*K*∈ T

*h*}, where

*p*>0 is an integer and

*Pp*(

*K*) denotes the space of all polynomials on

*K*of degree≤

*p*. A functionΦ

_{∈}S

*is, in general, discontinuous on interfacesΓ*

_{h}p_{∈ F}

*. ByΦ*

_{h}I*L*

_{Γ}and Φ

*R*

_{Γ}we denote the values ofΦonΓ considered from the interior and the exterior of

*KL*

Γ, respectively, and set

Φ_{Γ} = 1

2

Φ*L*_{Γ}+ Φ*R*_{Γ},
[Φ]Γ =ΦΓ*L*− Φ*R*Γ,

which denotes the average and jump ofΦonΓ.

The discrete problem is derived in the following way:
For arbitrary*t* ∈ [0,*T*] we can multiply the system by a
test functionS* _{h}p*, integrate over

*K*∈ T

*h, apply Green’s*the-orem, sum over all elements

*K*∈ T

*h, use the concept of*the numerical flux and introduce suitable terms mutually vanishing for a regular exact solution. Moreover, we carry out a suitable partial linearization of nonlinear terms and then we can define the following forms.

In order to evaluate the integrals overΓ ∈ F*h*in
invis-cid term we use the approximation

H(w*L*

Γ,w*R*Γ,*n*Γ)≈
2

*s*=1

f*s(*w)(*n*Γ)s,

whereHis a numerical flux. For the construction of the
numerical flux we use the properties (2) off*s. Let us define*
the matrix

P(w,*n*) :=

2

*s*=1

A*s(*w)*ns,*

where*n*=(*n*_{1},*n*2), *n*2_{1}+*n*2_{2}=1. Then we have

P(w,*n*)w:=

2

*s*=1

f*s(*w)*ns.*

It is possible to show that the matrixPis diagonalizable. It
means that there exists a nonsingular matrixT=T(w,*n*)
and a diagonal matrixΛ=Λ(w,*n*) such that

P=TΛT−1, Λ=diag(λ_{1}, ..., λ4),

whereλi=λi(w,*n*) are eigenvalues of the matrixP. Then
we can define the ”positive” and ”negative” parts of the
matrixPby

P±_{=}_{T}_{Λ}±_{T}−1_{,} _{Λ}±_{=}_{diag(λ}±
1, ..., λ±4),

whereλ+_{=}_{max(λ,}_{0),}_{λ}−_{=}_{min(λ,}_{0). Using this concept,}

we introduce the so-called Vijayasundaram numerical flux

H(w*L*_{,}_{w}*R*_{,}_{n}_{)}_{=}_{P}+

_{w}_{L}

+w*R*

2 ,*n*

w*L*_{+}_{P}−

_{w}_{L}

+w*R*

2 ,*n*

w*R*_{.}

This numerical flux has suitable form for a linearization. Now we can define inviscid form in the following way

*bh( ¯*w*h,*w*h, Φh) :*=

−

*K*∈T*h*

*K*

2

*s*=1

A*s( ¯*w*h)*w*h*·∂Φh
∂*xs* d*x*

+

Γ∈F*I*
*h*

Γ

P+_{(}_{}_{w}_{¯}

*h*,*n*Γ)w*Lh*+P−(w¯*h*,*n*Γ)w*Rh*

·[Φh] d*S*

+

Γ∈F*B*
*h*

Γ

P+_{(}

w¯*h*,*n*Γ)w*Lh*+P−(w¯*h*,*n*Γ) ¯w*Rh*

·Φhd*S*,

where the boundary state ¯w*R*

*h* is evaluated with the aid of
the local linearized Riemann problem described in [1]. For
the discretization of the viscous terms we use the property
(3) and get the viscous form

*ah( ¯*w*h,*w*h, Φh) :*=

+

*K*∈T*h*

*K*
2
*s*=1
2

*k*=1

K*s*,*k( ¯*w*h)* ∂w*h*

∂*xk* ·
∂Φh

∂*xs* d*x*

−
Γ∈F*I*
*h*

Γ 2

*s*=1

_{}2

*k*=1

K*s*,*k( ¯*w*h)*∂w*h*

∂*xk*

(*n*Γ)s·[Φh] d*S*

−
Γ∈F*B*

*h*
Γ
2
*s*=1
2
*k*=1

K*s*,*k( ¯*w*h)*

∂w*h*

∂*xk* (*n*Γ)s·Φhd*S*

−Θ

Γ∈F*I*
*h*

Γ 2

*s*=1

_{}2

*k*=1

K*T*
*k*,*s*( ¯w*h)*

∂Φh
∂*xk*

(*n*Γ)s·[w*h] dS*

−Θ

Γ∈F*B*
*h*

Γ 2

*s*=1
2

*k*=1

K*T*
*k*,*s*( ¯w*h)*

∂Φh

∂*xk* (*n*Γ)s·w*h*d*S*
We setΘ= 1 orΘ =0 orΘ= −1 and get the so-called

symmetric version (SIPG) or incomplete version (IIPG) or nonsymetric version (NIPG), respectively, of the dis-cretization of the viscous terms.

Further, we define the turbulent forms*ph,kh, the *
inte-rior and boundary penalty form*J*σ

*h* and the right-hand side
form*lh*in the following way:

*ph( ¯*w*h,*w*h, Φh) :*=

Γ∈F*I*
*h*

Γ 2

*s*=1

−
Γ∈F*I*
*h*

Γ 2

*s*=1

A*ps*( ¯w*Rh*)w*hR*·Φ*Rh* (*n*Γ)sd*S*

+

Γ∈F*B*
*h*

Γ 2

*s*=1

A*p*

*s*( ¯w*h)*w*h*·Φh(*n*Γ)sd*S*

−

*K*∈T*h*

*K*

2

*s*=1

A*ps*( ¯w*h)*w*h*·∂Φh
∂*xs* d*x*,

*kh( ¯*w*h, Φh) :*=

Γ∈F*I*
*h*

Γ 2

*s*=1

k*s( ¯*w*L*

*h*)·Φ*hL*(*n*Γ)sd*S*

−
Γ∈F*I*
*h*

Γ 2

*s*=1

k*s( ¯*w*R*

*h*)·Φ*Rh*(*n*Γ)sd*S*

+

Γ∈F*B*
*h*

Γ 2

*s*=1

k*s( ¯*w*h)*·Φh(*n*Γ)sd*S*

−

*K*∈T*h*

*K*

2

*s*=1

k*s( ¯*w*h)*·∂Φh_{∂}_{x}

*s* d*x*,

*J*σ

*h*(w*h, Φh) :*=

Γ∈F*I*
*h*

Γ

σ[w*h]*·[Φh] d*S*

+

Γ∈F*B*
*h*

Γ

σw*h*·Φhd*S*,

*lh( ¯*w*h, Φh) :*=

Γ∈F*B*
*h*

Γ

σw*B*·Φhd*S*

−Θ

Γ∈F*B*
*h*

Γ 2

*s*=1
2

*k*=1

K*T*
*k*,*s*( ¯w*h)*

∂Φh

∂*xk*(*n*Γ)s·w*B*d*S*,
whereσis a parameter of the method and boundary state

w*B*is defined on the basis of the Dirichlet boundary
condi-tions and extrapolation.

In the vicinity of discontinuities or steep gradients non-physical oscillations can appear in the approximate solu-tion. In order to overcome this difficulty we employ

artifi-cial viscosity forms, see [3]. They are based on the discon-tinuity indicator

g(*K*) := 1

*d*(*K*)|*K*|3/4

∂*K*[¯ρh]

2_{dS}_{,} _{K}_{∈ T}_{h,}

where [¯ρh] is the jump of the function ¯ρh(=the first

com-ponent of the vector function ¯w*h) on the boundary* ∂*K*,

*d*(*K*) denotes the diameter of*K* and|*K*| denotes the area
of the element*K*. Then we define the discrete
discontinu-ity indicator

*G*(*K*) :=0 ifg(*K*)<1,

*G*(*K*) :=1 ifg(*K*)≥1, *K*∈ T*h,*
and the artificial viscosity forms

˜

βh( ¯w*h,*w*h, Φh) :*=

ν1

*K*∈T*h*

*d*(*K*)*G*(*K*)

*K*∇w*h*· ∇Φhd*x*,

˜

*Jh( ¯*w*h,*w*h, Φh) :*=

ν2

Γ∈F*I*
*h*

1 2

*G*(*KL*

Γ)+*G*(*K*Γ*R*)
Γ

[w*h]*·[Φh] d*S*,

with constantsν1andν2.

All these forms are linear with respect tow*h*and
non-linear with respect to ¯w*h.*

Finally, we set

(ϕ, ψ)=

Ωh

ϕ ψd*x*.

**3.2 Full space-time DG discretization**

Let 0=*t*_{0}<*t*1< ... <*tM*=*T* be a partition of the interval
[0,*T*] and let us denote*Im*=(*tm*−1,*tm),*τm=*tm*−*tm*−1for
*m*=1, ...,*M*. We define the spaceS_{h}p_{,τ},*q*=(*S _{h}p*

_{,τ},

*q*)4, where

*Sp*,*q*
*h*,τ :=

φ;φ_{|}*Im* =

*q*

*i*=0

ζiφi, whereφi∈*S _{h}p*, ζi∈

*Pq*

_{(}

_{I}_{m)} .

with integers*p*,*q*≥1.*Pq*_{(}* _{I}_{m) denotes the space of all }*
poly-nomials in

*t*on

*Im*of degree≤

*q*. Moreover forΦ ∈ S

*,τ,*

_{h}p*q*

we introduce the following notation:

Φ±*m*=Φ(*t*±*m*)=* _{t}*lim

_{→}

_{t}*m*±Φ(*t*),

{Φ_{}}*m*=Φ+*m*− Φ−*m*.

Approximate solution w*h*τ of the problem will be sought

in the spaceS*p*,*q*

*h*,τ. Since the functions of this space are in

general discontinuous in time, we ensure the connection
between*Im*−1and*Im*by the penalty term in time

{w*h*τ}*m*−1, Φhτ(*t*+_{m}_{−}_{1})

.

The initial statew*h*τis included by the*L*2(Ωh(*t*0))-projection

ofw0_{on}_{S}*p*
*h*(*t*0):

w*h*τ(*t*_{0}+), Φhτ(*t*+_{0})

=w0, Φhτ(*t*+_{0})

∀Φhτ∈S_{h}p_{,τ},*q*.

Now we introduce a suitable linearization. We can use two possibilities.

1) We put ¯w*h*τ(*t*) :=w*h(t*−_{m}_{−}_{1}) for*t*∈*Im.*

2) We prolong the solution from the time interval*Im*−1

to the time interval*Im.*

We say that a functionw*h*τ ∈ S_{h}p_{,τ},*q* is the approximate

solution of the problem (1) obtained by the ST-DG method, if it satisfies the following conditions

*M*

*m*=1

*Im*

∂w*h*τ

∂*t* , Φhτ

d*t*

+

*M*

*m*=1

*Im*

(*ah( ¯*w*h*τ,w*h*τ, Φhτ)+*bh( ¯*w*h*τ,w*h*τ, Φhτ)) d*t*

+

*M*

*m*=1

*Im*

_{˜}

βh( ¯w*h*τ,w*h*τ, Φhτ)+*J*˜*h( ¯*w*h*τ,w*h*τ, Φhτ)

d*t*

+

*M*

*m*=1

*Im*

(*J*σ

−
Γ∈F*I*
*h*
Γ
2
*s*=1

A*ps*( ¯w*Rh*)w*hR*·Φ*Rh* (*n*Γ)sd*S*

+

Γ∈F*B*
*h*
Γ
2
*s*=1
A*p*

*s*( ¯w*h)*w*h*·Φh(*n*Γ)sd*S*

−

*K*∈T*h*

*K*

2

*s*=1

A*sp*( ¯w*h)*w*h*·∂Φh
∂*xs* d*x*,

*kh( ¯*w*h, Φh) :*=

Γ∈F*I*
*h*

Γ 2

*s*=1

k*s( ¯*w*L*

*h*)·Φ*hL*(*n*Γ)sd*S*

−
Γ∈F*I*
*h*
Γ
2
*s*=1

k*s( ¯*w*R*

*h*)·Φ*Rh* (*n*Γ)sd*S*

+

Γ∈F*B*
*h*

Γ 2

*s*=1

k*s( ¯*w*h)*·Φh(*n*Γ)sd*S*

−

*K*∈T*h*

*K*

2

*s*=1

k*s( ¯*w*h)*·∂Φh_{∂}_{x}

*s* d*x*,

*J*σ

*h*(w*h, Φh) :*=

Γ∈F*I*
*h*

Γ

σ[w*h]*·[Φh] d*S*

+

Γ∈F*B*
*h*

Γ

σw*h*·Φhd*S*,

*lh( ¯*w*h, Φh) :*=

Γ∈F*B*
*h*

Γ

σw*B*·Φhd*S*

−Θ

Γ∈F*B*
*h*
Γ
2
*s*=1
2
*k*=1
K*T*
*k*,*s*( ¯w*h)*

∂Φh

∂*xk*(*n*Γ)s·w*B*d*S*,
whereσis a parameter of the method and boundary state

w*B*is defined on the basis of the Dirichlet boundary
condi-tions and extrapolation.

In the vicinity of discontinuities or steep gradients non-physical oscillations can appear in the approximate solu-tion. In order to overcome this difficulty we employ

artifi-cial viscosity forms, see [3]. They are based on the discon-tinuity indicator

g(*K*) := 1

*d*(*K*)|*K*|3/4

∂*K*[¯ρh]

2_{dS}_{,} _{K}_{∈ T}_{h,}

where [¯ρh] is the jump of the function ¯ρh(=the first

com-ponent of the vector function ¯w*h) on the boundary* ∂*K*,

*d*(*K*) denotes the diameter of *K*and|*K*| denotes the area
of the element*K*. Then we define the discrete
discontinu-ity indicator

*G*(*K*) :=0 ifg(*K*)<1,

*G*(*K*) :=1 ifg(*K*)≥1, *K*∈ T*h,*
and the artificial viscosity forms

˜

βh( ¯w*h,*w*h, Φh) :*=

ν1

*K*∈T*h*

*d*(*K*)*G*(*K*)

*K*∇w*h*· ∇Φhd*x*,

˜

*Jh( ¯*w*h,*w*h, Φh) :*=

ν2

Γ∈F*I*
*h*

1 2

*G*(*KL*

Γ)+*G*(*K*Γ*R*)
Γ

[w*h]*·[Φh] d*S*,

with constantsν1andν2.

All these forms are linear with respect tow*h*and
non-linear with respect to ¯w*h.*

Finally, we set

(ϕ, ψ)=

Ωh

ϕ ψd*x*.

**3.2 Full space-time DG discretization**

Let 0=*t*_{0}<*t*1< ... <*tM*=*T* be a partition of the interval
[0,*T*] and let us denote*Im*=(*tm*−1,*tm),*τm=*tm*−*tm*−1for
*m*=1, ...,*M*. We define the spaceS_{h}p_{,τ},*q*=(*S _{h}p*

_{,τ},

*q*)4, where

*Sp*,*q*
*h*,τ :=

φ;φ_{|}*Im* =

*q*

*i*=0

ζiφi, whereφi∈*S _{h}p*, ζi∈

*Pq*

_{(}

_{I}_{m)} .

with integers*p*,*q*≥1.*Pq*_{(}* _{I}_{m) denotes the space of all }*
poly-nomials in

*t*on

*Im*of degree≤

*q*. Moreover forΦ ∈ S

*,τ,*

_{h}p*q*

we introduce the following notation:

Φ±*m*=Φ(*t*±*m*)=* _{t}*lim

_{→}

_{t}*m*±Φ(*t*),

{Φ_{}}*m*=Φ+*m*− Φ−*m*.

Approximate solution w*h*τ of the problem will be sought

in the spaceS*p*,*q*

*h*,τ. Since the functions of this space are in

general discontinuous in time, we ensure the connection
between*Im*−1and*Im*by the penalty term in time

{w*h*τ}*m*−1, Φhτ(*t*+_{m}_{−}_{1})

.

The initial statew*h*τis included by the*L*2(Ωh(*t*0))-projection

ofw0_{on}_{S}*p*
*h*(*t*0):

w*h*τ(*t*_{0}+), Φhτ(*t*+_{0})

=w0, Φhτ(*t*_{0}+)

∀Φhτ∈S* _{h}p*,

_{,τ}

*q*.

Now we introduce a suitable linearization. We can use two possibilities.

1) We put ¯w*h*τ(*t*) :=w*h(t*−_{m}_{−}_{1}) for*t*∈*Im.*

2) We prolong the solution from the time interval*Im*−1

to the time interval*Im.*

We say that a functionw*h*τ ∈ S* _{h}p*,

_{,τ}

*q*is the approximate

solution of the problem (1) obtained by the ST-DG method, if it satisfies the following conditions

*M*
*m*=1
*Im*

∂w*h*τ

∂*t* , Φhτ

d*t*
+
*M*
*m*=1
*Im*

(*ah( ¯*w*h*τ,w*h*τ, Φhτ)+*bh( ¯*w*h*τ,w*h*τ, Φhτ)) d*t*

+

*M*

*m*=1

*Im*

_{˜}

βh( ¯w*h*τ,w*h*τ, Φhτ)+*J*˜*h( ¯*w*h*τ,w*h*τ, Φhτ)

d*t*

+

*M*

*m*=1

*Im*

(*J*σ

*h*(w*h*τ, Φhτ)+*ph( ¯*w*h*τ,w*h*τ, Φhτ)) d*t*

+

*M*

*m*=2

{w*h*τ}*m*−1, Φhτ(*t*+*m*−1)

+ w*h*τ(*t*+_{0}), Φhτ(*t*_{0}+)

=
*M*
*m*=1
*Im*

*lh( ¯*w*h*τ, Φhτ) d*t*+

w0_{, Φh}
τ(*t*_{0}+)

+

*M*

*m*=1

*Im*

*kh( ¯*w*h*τ, Φhτ) d*t* ∀Φhτ∈S* _{h}p*,τ,

*q*.

**4 Discretization of the**

*k*

_{−}

### ω

**turbulence**

**model**

**4.1 Space discretization of the problem**

We apply discontinuous Galerkin method in the similar
way as in the previous section. Discretization is carried out
on the same mesh for simplicity. An approximate solution
of the problem is looked for in the space ˜S*p _{h}*˜ =(

*S*˜)2and a

_{h}pfunction ˜Φ_{∈}S˜*p _{h}*˜is, in general, discontinuous on interfaces
Γ

_{∈ F}

*.*

_{h}IIf we define ˜P± _{=}_{(}_{}_{ρ}_{v}_{ ·}_{n}_{Γ}_{)}±_{I}_{, where}_{I}_{=}_{diag(1,}_{1),}

and use the property (5) then we can define convect form ˜

*bh( ¯˜*w*h,*w˜*h,*Φh) :˜ =

−

*K*∈T*h*

*K*
2
*s*=1
˜

A*s*w˜*h*·∂
˜
Φh
∂*xs* d*x*

+

Γ∈F*I*
*h*

Γ

_{˜}

P+_{w}_{˜}*L*

*h*+P˜−w˜*Rh*

·[ ˜Φh] d*S*

+

Γ∈F*B*
*h*

Γ

_{˜}

P+_{w}_{˜}*L*

*h*+P˜−w¯˜*Rh*

·Φh˜ d*S*,

where the state ¯˜w*Rh* is based on the boundary conditions.
For the discretization of the diffusion term we use the

property (6) and then we have ˜

*ah( ¯˜*w*h,*w˜*h,*Φh) :˜ =

+

*K*∈T*h*

*K*

2

*s*=1

˜

K*s( ¯˜*w*h)*∂w˜*h*
∂*xs* ·

∂Φh˜
∂*xs* d*x*

−
Γ∈F*I*
*h*

Γ 2

*s*=1

˜

K*s( ¯˜*w*h)*∂w˜*h*
∂*xs*

(*n*Γ)s·[ ˜Φh] d*S*

−
Γ∈F*B*

*h*
Γ
2
*s*=1
˜

K*s( ¯˜*w*h)*∂w˜*h*
∂*xs*(*n*Γ)s·

˜
Φhd*S*

−Θ

Γ∈F*I*
*h*
Γ
2
*s*=1
˜

K*s( ¯˜*w*h)*∂Φh˜
∂*xs*

(*n*Γ)s·[ ˜w*h] dS*

−Θ

Γ∈F*B*
*h*

Γ 2

*s*=1

˜

K*s( ¯˜*w*h)*∂Φh˜

∂*xs*(*n*Γ)s·w˜*h*d*S*.

Further we define the turbulent form ˜*sh, the interior and*
boundary penalty form ˜*J*σ˜

*h* and the right-hand side form ˜*lh*
in the following way:

˜

*J*σ˜

*h*( ˜w*h,*Φh) :˜ =

Γ∈F*I*
*h*

Γ

˜

σ[ ˜w*h]*·[ ˜Φh] d*S*

+

Γ∈F*B*
*h*

Γ

˜

σw˜*h*·Φh˜ d*S*,

˜

*lh( ¯˜*w*h,*Φh) :˜ =

Γ∈F*B*
*h*

Γ

˜

σw˜*B*·Φh˜ d*S*

−Θ

Γ∈F*B*
*h*

Γ 2

*s*=1

˜

K*s( ¯˜*w*h)*∂Φh˜

∂*xs*(*n*Γ)s·w˜*B*d*S*.

˜

*sh( ¯˜*w*h,*Φh) :˜ =

*K*∈T*h*

*Ks*˜( ¯˜w*h)* ·
˜
Φhd*x*

Boundary state ˜w*B* is defined on the basis of the Dirichlet
boundary conditions and extrapolation.

**4.2 Full space-time DG discretization**

For the full space-time discontinuous Galerkin
discretiza-tion we use the same partidiscretiza-tion of the interval [0,*T*] as in
the previous section. We say that a function ˜w*h*τ ∈S˜* _{h}p*˜,˜

_{,τ}

*q*is

the approximate solution of the problem (4) obtained by the ST-DG method, if it satisfies the following conditions

*M*−1

*m*=0

*Im*

∂ρw˜*h*τ

∂*t* ,Φh˜ τ

d*t*

+

*M*−1

*m*=0
*Im*
˜

*ah( ¯˜*w*h*τ,w˜*h*τ,Φh˜ τ)+*b*˜*h( ¯˜*w*h*τ,w˜*h*τ,Φh˜ τ)

d*t*

+

*M*−1

*m*=0

*Im*

˜

*J*σ˜

*h*( ˜w*h*τ,Φh˜ τ) d*t*

+

*M*−1

*m*=1

{w˜*h*τ}*m,*Φh˜ τ(*t*+*m*)

+ w˜*h*τ(*t*_{0}+),Φh˜ τ(*t*+_{0})

=

*M*−1

*m*=0

*Im*

(˜*lh( ¯˜*w*h*τ,Φh˜ τ)+*s*˜*h( ¯˜*w*h*τ,Φh˜ τ)) d*t*

+w˜0,Φh˜ τ(*t*+_{0})

∀Φh˜ τ∈S˜* _{h}p*˜

_{,τ},˜

*q*.

**5 Numerical experiments**

In order to demonstrate the applicability of the developed
method we shall present two numerical simulations of
com-pressible turbulent flow in 2D. For this purpose we chose
two regimes for the profile RAE2822 and compared it with
experimental data, which can be found in [12]. First regime
is characterized by the far-field Mach number *M*∞ = 0.6

and the angle of the attackα=2.57*o*(’case 3’ in [12]) and

second case is characterized by the Mach number*M*∞ =

0.73 and the angle of the attack α = 3.19*o* (’case 9’ in

[12]). For both cases we set free-stream turbulence
inten-sity*Tu* =1%. Figures 1-3 and 5-7 show the pressure

Fig. 1.The distribution of the pressure for the case 3.

Fig. 2.The distribution of the Mach number for the case 3.

Fig. 3.The distribution of the turbulent kinetic energy for the case 3.

**6 Conclusion**

In this paper we dealt with the space-time discontinuous Galerkin method for the numerical solution of the viscous compressible turbulent flow. The applicability of the pro-posed method was demonstrated on the examples which show that the presented method is an efficient numerical

scheme for the solution of the viscous compressible turbu-lent flow.

Fig. 4.Comparison of the computed pressure coefficient*cp*with

the experimental data for the case 3.

Fig. 5.The distribution of the pressure for the case 9.

Fig. 6.The distribution of the Mach number for the case 9.

**Acknowledgment**

Fig. 1.The distribution of the pressure for the case 3.

Fig. 2.The distribution of the Mach number for the case 3.

Fig. 3.The distribution of the turbulent kinetic energy for the case 3.

**6 Conclusion**

In this paper we dealt with the space-time discontinuous Galerkin method for the numerical solution of the viscous compressible turbulent flow. The applicability of the pro-posed method was demonstrated on the examples which show that the presented method is an efficient numerical

scheme for the solution of the viscous compressible turbu-lent flow.

Fig. 4.Comparison of the computed pressure coefficient*cp*with

the experimental data for the case 3.

Fig. 5.The distribution of the pressure for the case 9.

Fig. 6.The distribution of the Mach number for the case 9.

**Acknowledgment**

This result originated with the support of Ministry of In-dustry and Trade of the Czech Republic for the long-term strategic development of the research organization. The au-thor acknowledge this support.

Fig. 7.The distribution of the turbulent kinetic energy for the case 9.

Fig. 8.Comparison of the computed pressure coefficient*cp*with

the experimental data for the case 9.

Fig. 9.Detail of the mesh.

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*Mathematical and Computational Methods for *
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4. J. ˇCesenek, et al., *Appl.* *Math.* *Comput.*,
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