8/10/2019 Lecture5 CFD
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8/10/2019 Lecture5 CFD
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From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Outline of lecture 5
From conservation laws to Navier-Stokes equations,
Finite-Volume formulation of Navier-Stokes equations,
Mass and Momentum equations,
Volume and surface integral evaluation
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
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3/29
From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Outline of lecture 5
From conservation laws to Navier-Stokes equations,
Finite-Volume formulation of Navier-Stokes equations,
Mass and Momentum equations,
Volume and surface integral evaluation
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
http://find/8/10/2019 Lecture5 CFD
4/29
From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Outline of lecture 5
From conservation laws to Navier-Stokes equations,
Finite-Volume formulation of Navier-Stokes equations,
Mass and Momentum equations,
Volume and surface integral evaluation
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
http://find/http://goback/8/10/2019 Lecture5 CFD
5/29
From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Outline of lecture 5
From conservation laws to Navier-Stokes equations,
Finite-Volume formulation of Navier-Stokes equations,
Mass and Momentum equations,
Volume and surface integral evaluation
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
F i l N i S k i
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From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Derivation according to an arbitrary velocity field
Derivation according to an arbitrary velocity field
Let
Ud be an arbitrary velocity field and g(x, t) a scalar function.
g
t=g
t+g
Ud
For a domain Ddwith a boundaryDdmoving with the velocity
Ud,
we have :
t
DdgdV=
Dd
g
t dV+
Ddg
Ud
n dS
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
F ti l t N i St k ti
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From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Derivation according to an arbitrary velocity field
This relation can be interpreted as follows :
Variation of the inte-
gral ofgin the moving
domain Dd
=
Integral of the tem-
poral variation of g
on Dd
+
Convective flux
convectif of g
across the bound-
aryD
d
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier Stokes equations
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From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Derivation according to an arbitrary velocity field
Let us recall the theorem establishing the expression of the material
derivative of a volume integral.Let Dmbe a material domain. With some regularity hypothesis on the
fields, we have :
d
dt
Dm
gdV= Dm
g
t
dV+Dm
gU
n dS (1)
For the material domain Dmcoinciding with Ddat timet, we get the
following relation :
d
dt
Dm
g dV=
t
Dd
g dV+
Dd
g
U
Udn dS (2)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
RANSE formulation
Reynolds Averaged Navier-Stokes Equations
Mass conservation
For isothermal viscous flows, conservation laws are reduced to
momentum and mass conservation. Let us consider a domain Dd ofboundaryDdmoving with the velocity
Ud and Dm, a material domain
coinciding with the material domain at time t.
Mass conservation :d
dt
DmdV= 0
Momentum conservation :
d
dt
D
m
U dV=
D
m
fv dV+
D
m
T dS
fv : volumic force (for us, gravity)T : Surface constraint
T =
n ,
: constraint tensor
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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From conservation laws to Navier Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
RANSE formulation
Reynolds Averaged Navier-Stokes Equations
Mass conservation
For a Newtonian fluid:
= (p+2
3div(
U))I+ 2D
wherepis the pressure, the dynamic viscosity, I the identity tensor
and D the rate-of-strain tensor.
Using this relation leads to:
d
dt
Dm
U dV =
Dm
fv dV
Dm
p+
2
3div(
U)
n dS+
Dm
2Dn dS
(3)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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From conservation laws to Navier Stokes equations
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
RANSE formulation
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Using relation (2) and the Gauss theorem, we get the new expression
of the mass and momentum conservation equations:
t
Dd
dV+
Dd
(U
Ud) n
dS= 0 (4)
t
Dd
U dV+
Dd
U
(U
Ud) n
dS=
Dd
fv dV
Dd
p+
2
3div(
U)
dV +
Dd
div(2D) dV
(5)
wherefv includes the gravity
g and additional source terms to be
specified later.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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q
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Notations
Let us consider an incompressible and isothermal flow of a fluid
defined by its density and dynamic viscosity . LetUibe the
cartesian components in directionsiof velocity,fithe components ofthe volume forces andpthe pressure.
Momentum and mass conservation laws may be written under an
integral formulation for a volume Vbounded by surfaces S, moving at
velocity
Ud.n is an outbound normalized normal-to-the-face vector.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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q
Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
From equations (4)and (5), one gets :
t
Dd
dV+
Dd
(U
Ud) n
dS= 0 (6)
t
Dd
UidV +
Dd
Ui
(U
Ud) n
dS=
Dd
fvidV
D
d
xi
p+
2
3div(
U)
dV +
D
d
2
xj
DijdV
(7)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
RANSE formulation
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations RANSE formulation
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Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Mean and fluctuating parts decomposition 1/3
The instantaneous velocity components and pressure are
decomposed into mean and fluctuating parts:
Ui = Ui + u
i
p = p+ p (8)
which leads to the modified mass and momentum conservation
equations (also called Reynolds Averaged Navier-Stokes Equations(RANSE)).
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations
Fi i V l f l i f N i S k i
RANSE formulation
R ld A d N i S k E i
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Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Mean and fluctuating parts decomposition 2/3
One gets the mean mass and momentum conservation equations:
t
Dd
dV+
Dd
(UjUd
j )njdS= 0 (9)
t
Dd
UidV+
Dd
Ui
(UjUd
j )nj
dS+
Dd
uiu
jnjdS=
Dd
fvi dV
Dd
xip
+
2
3div
(
U)dV
+
Dd
xj
2Dij
dV
(10)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations
Fi it V l f l ti f N i St k ti
RANSE formulation
R ld A d N i St k E ti
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Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Mean and fluctuating parts decomposition 3/3
The Reynolds stress tensor uiu
jis defined by :
uiu
j= 2
3kij + 2tDij + f
EASM (11)
whereij is the Kronecker symbol.
Terms t,kandfEASM are the turbulent viscosity coefficient, turbulent
kinetic energy and source terms associated with the non-linear EASM
turbulence closure, respectively. For isotropic linear turbulenceclosures,fEASM = 0.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier Stokes equations
RANSE formulation
Reynolds Averaged Navier Stokes Equations
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Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Reynolds Averaged Navier-Stokes Equations
Once the definition of the Reynolds stresses is included in theequations, one finally gets :
t
D
d
dV+
D
d
(UjUd
j )njdS= 0 (12)
t
Dd
UidV+
Dd
Ui
(UjUd
j )nj
dS=
Dd
(fvi + fEASMi )dV
Dd
pT
xi
dV+
Dd
xj2effDijdV
(13)
with:
pT = p+2
3div(
U) +2k
3eff = +t
(14)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equations
Finite Volume formulation of Navier-Stokes equations
RANSE formulation
Reynolds Averaged Navier-Stokes Equations
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Finite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Reynolds Averaged Navier-Stokes Equations
Mass conservation
Mass conservation for a fluid with uniform density
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
RANSE formulationReynolds Averaged Navier-Stokes Equations
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Finite Volume formulation of Navier Stokes equations
Discretization of RANSE
Reynolds Averaged Navier Stokes Equations
Mass conservation
If the density is uniform and constant, then the mass conservation
equation becomes:
t
V
dV+
S(U
Ud) n dS= 0 (15)
The displacement of a surface limiting a volume control should verify
the following spatial conservation formula (geometrical identity) :
t
V
dV
S
Ud
n dS= 0 (16)
And, then, even for a moving domain, mass conservation equation
reads as follows :
S
U
n dS= 0 (17)
which gives, once discretized :
f
Uf
Sf= 0 (18)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
RANSE formulationReynolds Averaged Navier-Stokes Equations
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Finite Volume formulation of Navier Stokes equations
Discretization of RANSE
Reynolds Averaged Navier Stokes Equations
Mass conservation
Reynolds Averaged Navier-Stokes Equations 1/2
This leads to further simplifications of the mean momentum equations.
xj(2effDij) =
xj( +t)(Ui
xj+Uj
xi)
=
xj
( +t)(
Ui
xj)
+t
xj
Uj
xi
(19)
where the reduced incompressibility condition div(
U) = 0 has beenused to simplify this formulation.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
RANSE formulationReynolds Averaged Navier-Stokes Equations
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q
Discretization of RANSE
y g q
Mass conservation
Reynolds Averaged Navier-Stokes Equations 2/2
Using the incompressibility condition and Gauss theorem leads to the
final formulation of the momentum conservation :
t
Dd
UidV+Dd
Ui(UjUdj )njdS= Dd
(fvi + fEASMi )dV
Dd
pTnidS+
Dd
eff
Ui
xj
njdS+
Dd
t
xj
Uj
xidV
(20)
with:
pT = p+2k
3eff = +t
(21)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Finite volume discretisationEvaluation of volume integrals
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q
Discretization of RANSE
g
Evaluation of surface integrals
Principles of finite volume discretisation
The computational domain is discretized with an unstructured grid and
each individual cell volumeVis considered as a control volume where
the integral formulation of the Navier-Stokes equations has to be
satisfied.
All variables are located at the cell geometric centers of control volume
V andno hypothesis is made concerning the shape of this control
volumei.e. a control volume is made of an arbitrary number of
constitutive faces noted S.This peculiarity is fundamental if one wants to implement local mesh
adaption strategies later on.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Finite volume discretisationEvaluation of volume integrals
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Discretization of RANSE Evaluation of surface integrals
Principles of finite volume discretisation
Typical unstructured control volume
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Finite volume discretisationEvaluation of volume integrals
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Discretization of RANSE Evaluation of surface integrals
Volume integrals
If we postulate a spatial linear variation of Qleading to a second order
discretisation in space:
Q(x) = QP + (xxP) (Q)P (22)
withQP= Q(xP). The integral of a functionQover a domain V isapproximated by:
V
QdV =
V(QP + (xxP) (Q)P)dV
= QP
VdV+
V
(xxP) (Q)P)dV
= QPV
(23)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Di i i f RANSE
Finite volume discretisationEvaluation of volume integrals
E l i f f i l
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Discretization of RANSE Evaluation of surface integrals
Surface integrals
The surface integral splits into a sum of faces of the discrete volume ofintegrationV:
S
QndS = f
S
nQfdSf
= f
Sn(Qf + (xxP) (Q)f)dSf
= f
SfQf
(24)
where
Sf =SndSf is the surface-oriented vector andQfis the valueofQat the center of the face. All variablesQbeing located at the
center of cells, one has first to rebuild the value of function at the
center of the face (noted here Qf) from the cell-centered values of the
function (QL etQR) from each side of the face.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Di ti ti f RANSE
Finite volume discretisationEvaluation of volume integrals
E l ti f f i t l
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Discretization of RANSE Evaluation of surface integrals
Discretization of the momentum equations
Transport equation for a generic variable Qfor a cellV
of centerCand limited by an arbitrary number of faces fis given by :
(VQ)C +
t(VQ)C +
f
(FcfFdf) = (SVQ ) +
f
(SfQ) (25)
Fcf =
SQ
(UjU
dj )nj
dS
= .
mfQf
Fdf =
SeffQ
xjnjdS
.mf = (UjUd
j )njSf
(26)
Terms Fcf and Fdfare respectively convection and diffusion fluxes
across the facef,.
mfbeing the mass flux across this face. SVQ andS
fQ
are surface and volume source terms.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Finite volume discretisationEvaluation of volume integrals
Evaluation of surface integrals
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Discretization of RANSE Evaluation of surface integrals
Discretization of the momentum equations
For momentum equations,Q = Uiand the pressure term associatedto the i momentum equation will be included in the surface source term
as:
SfQ =
SpTnidS (27)
Temporal derivatives are evaluated by upwind second-order
discretisation :A
t ecAc + epAp+ eqAq (28)
Subscriptcstands for the current time step, and petqrefer toprevious time steps.
Remark These coefficients will be notedeq,ep,ec oreq,ep,ec inthese lectures.
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
From conservation laws to Navier-Stokes equationsFinite Volume formulation of Navier-Stokes equations
Discretization of RANSE
Finite volume discretisationEvaluation of volume integrals
Evaluation of surface integrals
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Discretization of RANSE Evaluation of surface integrals
The first term of the left-hand side of equation (25) corresponds to a
pseudo-steady term needed to stabilize the solution procedure forsteady flows. The corresponding derivative is evaluated by :
A
= (AcAc0)/ (29)
Ac0 is the previous estimation ofAc within the non-linear loop. Finally,
a generic discrete transport equation reads :
(ec + 1/)(VQ)cC +f
(FcfFdf) = (SVQ ) +
f
(SfQ)
(eVQ)pC (eVQ)qC + (VQ)
c0C/
(30)
ECN-CNRS A generalized unstructured finite volume discretisation - Part 1
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