CFD – AN OVERVIEW

H. Klimach, harald.klimach@dlr.de
Institute of Software Methods for Product Virtualization (SP), DLR
 Computational

§ Complex problems

§ No analytical solution

§ Numerical approximation

§ Using computers

2 02-Basics_CFD
 Fluid

§ Continually deforms when force is applied

Images: game-icons.net, Lorc and Delapouite

3 02-Basics_CFD
 Dynamics

§ Considering the forces in fluids

§ Studying the resulting fluid motion

§ May be stationary or instationary

4 02-Basics_CFD
 Classes of Fluids

§ Liquids
§ Forms free surface

§ Gases
§ Freely diffuse

§ Plasma
§ Ionized, highly electrical conductive

Images: game-icons.net, Lorc and Delapouite

5 02-Basics_CFD
 Some Defining Properties of Fluids

§ Viscosity
§ Relation of shear stress to speed of deformation
§ Compressibility
§ Relation of volume change under pressure
§ Heat conductivity

§ Density

Images: game-icons.net, Cathelineau, Skoll, Lorc and Delapouite

6 02-Basics_CFD
 Modelling

Reality

Physical Model

Mathematical Model

Numerical Approximation

Simulation code

Usage of the simulation code

Analysis of results

7 02-Basics_CFD
 Physical Model for Fluids

§ Level of detail
§ Quantum – Molecular – Continuum

§ Consideration of effects
§ Electrodynamics? Gravity? Relativity? Chemical reactions?

8 02-Basics_CFD
 Continuum Assumption

§ On the considered scale:

§ Fluid covers complete space
§ Infinitisimal small decomposition possible

§ Knudsen number:

mean free path

Kn =
reference length scale

9 02-Basics_CFD
 Continuum Classification

§ High Kn (>0.5):
§ (For example rarefied gases)
§ Physical model: Kinetic gas theory
§ Knudsen flow (0.01 < Kn < 0.5)

§ Low Kn (<0.01):
§ (For example air around plane)
§ Physical model: Continuum mechanics

Image: game-icons.net, Lorc

10 02-Basics_CFD
 Physical Continuum Model

§ Newtonian mechanics

§ Conservation laws:
§ Conservation of mass
§ Conservation of momentum
§ Conservation of energy

11 02-Basics_CFD
 Mathematical Model

§ Conservation laws yield compressible Navier-Stokes equations

§ System of partial differential equations

§ Describes evolution of state due to spatial fluid variations

12 02-Basics_CFD
 Conservation Laws

§ Quantity can neither vanish nor appear

§ Observing a given volume:
§ Quantity only changed by transport through the surface

§ Mathematical: @A
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~n
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<latexit

A
Z I
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<latexit

@
u(~x, t)dA = f~(u)~nds
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@t A @A

13 02-Basics_CFD
 Integral Conservation

Z I
@
u(~x, t)dA = f~(u) · ~nds
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@t A @A

time t <latexit sha1_base64="Whf6eB2P1u39HRrcEMP75DgjaZo=">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</latexit>

<latexit

volume/area A <latexit sha1_base64="PgW2ZYpeAi+iQXQqE7VSXvYchk4=">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</latexit>

<latexit

location ~x <latexit sha1_base64="dXfbi5v/om81HZSSTDiowI3nSW4=">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</latexit>

<latexit
surface @A
<latexit sha1_base64="08Nfkh0MR8PJwZYTgLeA/37bxbs=">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</latexit>

conserved quantity u <latexit sha1_base64="PBsR5Uk4VHE3K1Wei2UYHOBp8lY=">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</latexit>

<latexit
surface normal ~n <latexit sha1_base64="Eq7Ci93vl5JRywtTO6WYLfGClQo=">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</latexit>
<latexit

flux
f~
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14 02-Basics_CFD
 Convective Transport

§ Quantity transported by velocity ~v

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<latexit

§ Convective flux: f~ = u · ~v
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15 02-Basics_CFD
 Conservation in Continuum

§ Conservation law holds for infinitisimal volumes

§ Conservation in each point
Ø Differential formulation

• Continuously differentiable:
• Divergence Theorem (replace surface by volume integral)
• Leibniz Rule (exchange integration and differentiation)

16 02-Basics_CFD
 Replacing Surface by Volume Integral

§ Divergence Theorem:
I Z
f~ · ~nds = r · f~dA
@A <latexit sha1_base64="zZuKSP3PK6dQurFiZUvCVyVB5xw=">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</latexit>
A
§ Incoporate in the conservation law
Z I
@
u(~x, t)dA = f~(u) · ~nds
<latexit sha1_base64="j3kCWPCeJGXE62iw5O4znMsWRQk=">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</latexit>
@t A @A

Z Z
§ Yields: @
u(~x, t)dA = r · f~(u)dA
<latexit sha1_base64="9MqeYEZvNCp84qG37vfStQsbuaA=">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</latexit>
@t A A
17 02-Basics_CFD
 Differential Conservation Law

§ Using Leibniz's rule and pulling everything into one integral:

Z ✓ ◆
@u(~x, t)
+ r · f~(u) dA = 0
<latexit sha1_base64="MzPdq84rDAnfxkNjl5FLTCX4xi0=">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</latexit>
A @t

§ Conservation holds for any volume

Ø Conservation in each point

@u(~x, t)
+ r · f~(u) = 0
<latexit sha1_base64="F9aQvZMm684jBAqUA+pn4ZElZeY=">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</latexit>
@t
18 02-Basics_CFD
 Conservation of Mass

§ Conserved: Mass density ⇢

<latexit sha1_base64="2VVbe9OvnocbBJaY+KdA4G5dZmg=">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</latexit>

§ Changed by:
§ Convection f~ = ⇢~v
<latexit sha1_base64="pQ85aWSlg2t7+I0nm6EnhkGpIZ0=">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</latexit>

Z I
@
§ Integral form: ⇢(~x, t)dA = ⇢~v · ~nds
<latexit sha1_base64="ZfF3d8RxMkioXr94/t0BvEej6Ls=">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</latexit>
@t A @A

@⇢(~x, t)
§ Differential form: + r · (⇢~v ) = 0
<latexit sha1_base64="oxXpGUn0yjzKWV+C1UkZ2EgcPxs=">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</latexit>
@t
19 02-Basics_CFD
 Conservation of Momentum

§ Conserved: Momentum density m

~ = ⇢~v
<latexit sha1_base64="iXSJnWB0WSLA9h4B6B+ZkMi6hdA=">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</latexit>

§ Changed by:

§ Convection
<latexit sha1_base64="VJqePWp/i8emM1OE1f2yhMcTut0=">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</latexit>
fc = m
~ ~v (Dyadic product)

§ Surface forces

§ pressure <latexit sha1_base64="9MSHD+dgRVHsQ7KVAbetj6vf8YY=">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</latexit>

f p = pI (Identity tensor)

§ friction
<latexit sha1_base64="lvsXcN9dSNlyr80WxZOS6B5XbkE=">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</latexit>
fv = ⌧ (Stress deviator tensor)

§ Possible volume forces (gravity, electromagnetism) F~ e

<latexit sha1_base64="BfNl1uD+Emd5/xE4WXiFze8Qiaw=">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</latexit>

20 02-Basics_CFD
 Conservation of Momentum

§ Integral form:
pres. fri.
Z I ⇣
convect.
⌘ Z
@
m(~
~ x, t)dA = m
~ ~v + pI ⌧ ~nds + F~ e dA
<latexit sha1_base64="ZPbYC2WH0+ZFFx6rXUL3Q93ideg=">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</latexit>
@t A @A A
§ Differential form:

convect. pres. fric.

@ m(~
~ x, t)
~ ~v + pI) = r · ⌧ + F~ e
+ r · (m
<latexit sha1_base64="8egCZ387UNG/pQHMxeeSa6yD4E4=">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</latexit>
@t

21 02-Basics_CFD
 Stress Deviator Tensor

@vj @vi
§ Deformation tensor: (def ~v )ij = +
@xi @xj
<latexit sha1_base64="TZiVHrXYterEWILoWQytOGeUdRI=">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</latexit>

§ Stress tensor in Navier-Stokes equations:

<latexit sha1_base64="+yWBNmpEaKJutJurEj5oUlYc3Qs=">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</latexit>
⌧= (r · ~v ) I + µ · def ~v
volume viscosity <latexit sha1_base64="PIHiYjtRbyl3D43sABfVUEC44Wo=">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</latexit>

dynamic shear viscosity µ

<latexit sha1_base64="e5Qhbnbiq3ISDZRXSTPE4soANOc=">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</latexit>

22 02-Basics_CFD
 Newtonian Fluids

§ Newtonian fluid:
§ Viscosity independent of shear rate
§ Still may depend on thermodynamic quantities (often temperature)

§ Non-Newtonian fluids:
§ Viscosity varies with shear rate
§ Rheology

23 02-Basics_CFD
 Conservation of Energy

⇢
§ Conserved: Energy density e = ⇢" + ~v · ~v +⇢gh
2
<latexit sha1_base64="4pfnvAV7vtsiB9V/bfAoalC5uiw=">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</latexit>
<latexit sha1_base64="3pMCeBW0qhBVxGGXqfv0YbDLKS4=">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</latexit>

§ Changed by:
§ Convection f~c = e · ~v
<latexit sha1_base64="WzVKHjfjf3kn8dlxRgzK1XHc128=">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</latexit>

§ Work of surface forces

f~s = p · ~v ⌧~v
§ Heat conduction
<latexit sha1_base64="jSUjI23YN1LpsU6jllfFpV4/aRg=">AAADB3ichVHLTttAFD2Y97MBlt1YREjdEDkIibKIhCigbpBAIgWJABpPJsGKY1v2JBKN/AP8Rn+gq6pblt2WFXwLC85MDeKhirHG98655565d66fhEGmPe92yBkeGR0bn5icmp6ZnftQml/4lsW9VKq6jMM4PfZFpsIgUnUd6FAdJ6kSXT9UR37ni4kf9VWaBXF0qC8TddoV7ShoBVJoQuelWqOv5KCVn2VuzU0ashlri/Rzd8VtxEw1yoNnnha9PC8456WyV/Hsct861cIpo1j7cekODTQRQ6KHLhQiaPohBDJ+J6jCQ0LsFANiKb3AxhVyTDG3R5YiQxDt8N/m6aRAI56NZmazJW8JuVNmuljm3rWKPtnmVkU/o73n/m6x9n9vGFhlU+ElrU/FSau4R1zjgoz3MrsF87GW9zNNVxotfLbdBKwvsYjpUz7pbDOSEuvYiIsdy2xTw7fnPl8goq2zAvPKjwqu7bhJK6xVViUqFAX1Ulrz+qyHY66+Hupbp75a2ah4B2vlza1i3hP4iCV84lDXsYmv2GcZEj/wB39x41w5P51fzu9/VGeoyFnEi+VcPwDRyak8</latexit>

f~h = rT
<latexit sha1_base64="E3omCHUqin0LLY0WeQoMP4kZIEU=">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</latexit>

§ Possible work by volume forces; Heat sources (e.g. chemical reactions,

radiation) e
F~ · ~v + Q
<latexit sha1_base64="A5XuKZB3qB3crcd9duTbBIjX2Jk=">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</latexit>

24 02-Basics_CFD
 Conservation of Energy

§ Integral form:

Z I conv pres. fric. heat Z

@ .
e(~x, t)dA = (e · ~v + p · ~v ⌧~v rT )~nds + (F~ e · ~v + Q)dA
<latexit sha1_base64="kX7/hO/F7/LJ+WZ4mArj9oKMCsU=">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</latexit>
@t A @A A

§ Differential form:

@e(~x, t)
+ r(e + p)~v = r · ⌧~v + r · (rT ) + F~ e · ~v + Q
<latexit sha1_base64="kCx+ycdvWqnGqE4PXSbZr8seTpI=">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</latexit>
@t

25 02-Basics_CFD
 Equation of State

§ Relation of thermodynamic quantities density, pressure and temperature

§ Simplest thermodynamic model is the ideal gas assumption:

p
=R·T
<latexit sha1_base64="dQj/6lPy/oWqEB7XkJmP5lEiHSU=">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</latexit>
⇢
gas constant

26 02-Basics_CFD
 Calorically Perfect Gas

§ Specific heat capacity assumed to be constant

cv
§ Direct proportional relation between inner energy and temperature:
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§ Together with the ideal gas model:

" = cv · T
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p
"= with as ratio
⇢( 1)
<latexit sha1_base64="I1qkR3bW80V4Tti5nHlWFwuIess=">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</latexit>

of specific
<latexit sha1_base64="3yup0ZTg3Y8cPel8azHrt9cxnaM=">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</latexit>

heats

27 02-Basics_CFD
 Real Gas

§ Usually ideal gas sufficient

§ More complex models required for

§ close to phase changes
§ close to critical points
§ high pressures
§ low temperatures
§ Compressibility factor as measure

28 02-Basics_CFD
 Real Gas Models

§ Virial model: Series of perturbative terms

§ Van der Waals: Most prominent 2 term model

§ Redlich-Kwong: 2 term model often more accurate than van der Waals

§ Many more of varying complexity

29 02-Basics_CFD
 Compressible Navier-Stokes Equations

@⇢(~x, t)
+ r · (⇢~v ) = 0
<latexit sha1_base64="I/v/rYzbRvQcBiWjR59nNn3x5MQ=">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</latexit>
@t
@ m(~
~ x, t)
+ r · (m~ ~v + pI) = r · ⌧
<latexit sha1_base64="Ls/XFY5rmRVMBqEU0IioS4xjM8U=">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</latexit>
@t
@e(~x, t)
+ r · (e + p)~v = r · ⌧~v + r · (rT )
@t
<latexit sha1_base64="NAVN81d/WWE6cw4xo55oS7Z/cE8=">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</latexit>

+Equation of state
⇣ ⇢ 2 ⌘
- Perfect gas: p=( 1) · e |~v |
<latexit sha1_base64="862CZ2La2n//u0GmF+mfhJ8AU6U=">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</latexit>
2 Image: game-icons.net, Lorc

30 02-Basics_CFD
 Nondimensionalization

§ Introduce suitable reference values

§ Define all quantities in relation to reference values

§ Useful for classification of flows

§ Helps to limit value range

31 02-Basics_CFD
 Reference Values

§ Choose a characteristic length, pressure, density and velocity

L, pref , ⇢ref , vref
<latexit sha1_base64="6dM+ytWP+QNumT2l793bIY4Q2P4=">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</latexit>

§ Accordingly you get a characteristic time scale, speed of sound and

temperature
r
L pref pref
tref = , cref = , Tref =
<latexit sha1_base64="v4nSApn3KgGcNUdozHgfd8iyLE4=">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</latexit>
vref ⇢ref R⇢ref
§ Normalize all quantities with the references

32 02-Basics_CFD
 Similarity Parameters

§ Ratios of flow effects, experiments with the same parameters show same
behavior

v Flow velocity
§ Mach number Ma =
c <latexit sha1_base64="rTSLhJeiZm7wLmLiPaVGzg47UnQ=">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</latexit>
Speed of sound

⇢vL Fictitious force

§ Reynolds number Re = Friction
µ <latexit sha1_base64="DPOeNlxaJ3+EaiXOkH+rtw3SRac=">AAAC43ichVHLSsNAFD3GV+uz6krcBIvgqqQiqAtBfOFCQcWqYEUmcdqG5sUkLWgp/oArcevSrX6PfosLT8YoqIgTJvfOueeeuXeuHXlunFjWS4/R29c/MJjLDw2PjI6NFyYmj+OwpRxZcUIvVKe2iKXnBrKSuIknTyMlhW978sRubqTxk7ZUsRsGR8lVJM99UQ/cmuuIhNBFYfpQmqtmtaaE06mqRmi2zd1up+q3uheFolWy9DJ/O+XMKSJb+2HhFVVcIoSDFnxIBEjoexCI+Z2hDAsRsXN0iCl6ro5LdDHE3BZZkgxBtMl/naezDA14TjVjne3wFo9bMdPEHPe2VrTJTm+V9GPaN+5rjdX/vKGjldMKr2htKua14h7xBA0y/sv0M+ZnLf9npl0lqGFZd+OyvkgjaZ/Ol84mI4pYU0dMbGlmnRq2Prf5AgFthRWkr/ypYOqOL2mFtlKrBJmioJ6iTV+f9XDM5Z9D/e1UFkorpfLBYnFtPZt3DjOYxTyHuoQ17GCfZTi4wSOe8GzUjFvjzrj/oBo9Wc4Uvi3j4R1vQZiH</latexit>

§ Prandtl number
µcp Momentum diff.
Pr = Thermal diff.

<latexit sha1_base64="AA9Woflz6aac7AmX2awXRddnqvM=">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</latexit>

33 02-Basics_CFD
 Nondimensional Navier-Stokes Equations

§ Similarity parameters of reference, perfect gas:

@⇢(~x, t)
+ r · (⇢~v ) = 0
@t <latexit sha1_base64="rIqKqDdB5/3K6TvRh3MndGQBJak=">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</latexit>

@ m(~
~ x, t) p r·⌧
+ r · (m
~ ~v + 2
I) =
@t Ma Re
✓ ◆
<latexit sha1_base64="bsiy69AHna5A1y0LsIkvNlv5uw8=">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</latexit>

@e(~x, t) p 1 T
+ r · (e + )~v = r · ⌧~v +
<latexit sha1_base64="Bcd6OwoFn4HxIF5UA6hBsuZddXA=">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</latexit>
@t M a2 Re ( 1)M a2 · P r
⇣ ⇢ 2⌘
p=( 1) · e |~v | M a2
<latexit sha1_base64="lxkHshMg7oSze4dmpE3hHgteDiY=">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</latexit>
2 Image: game-icons.net, Lorc

34 02-Basics_CFD
 Other Similarity Parameters

§ Depending on the problem

§ Some examples:
§ Péclet number Pe (transport)
§ Froude number Fr (gravitation)
§ Richardson number Ri (weather)
§ Rayleigh number Ra (free convection)
§ Strouhal number St (vortex shedding)

35 02-Basics_CFD
 Simplifications

§ Two important simplifications often deployed:

§ Inviscous flows: Euler equations

§ Neglecting diffusive processes

§ Incompressible flows
§ Density independent of pressure
§ Infinite speed of sound

36 02-Basics_CFD
 Simplification Overview

Compressible Navier-Stokes equations

include friction and thermal conduction

hyperbolic – parabolic

Re ! 1
<latexit sha1_base64="I5luM6vLngbZrl954Tr/g73LDLE=">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</latexit>
Ma ! 0
<latexit sha1_base64="HqS3/c/EXuJETRyDySrCTRr3Ki8=">AAAC2nichVFLS8NAEB7jq/UZ9eglWARPJRVBvRVfeCkoGC20pWzimi7Ni81W0eLJk3j16FX/k/4WD35ZU0GLdMNmZr/55tuZHTcJRKps+33MGJ+YnJouFGdm5+YXFs2l5fM07kmPO14cxLLuspQHIuKOEirg9URyFroBv3C7+1n84prLVMTRmbpNeCtkfiSuhMcUoLZp1pjVlMLvKCZlfGPZbbNkl229rGGnkjslytdJbH5Qky4pJo96FBKniBT8gBil+BpUIZsSYC3qA5PwhI5zuqcZ5PbA4mAwoF38fZwaORrhnGmmOtvDLQG2RKZF69hHWtEFO7uVw09hP7HvNOb/e0NfK2cV3sK6UCxqxRpwRR0wRmWGOXNQy+jMrCtFV7SjuxGoL9FI1qf3o3OAiATW1RGLDjXTh4arz9d4gQjWQQXZKw8ULN3xJSzTlmuVKFdk0JOw2eujHoy58neow46zWd4tV063StW9fN4FWqU12sBQt6lKx3SCMjyU9EKv9Ga0jAfj0Xj6phpjec4K/VrG8xccIpTv</latexit>

Incompressible Navier-
Euler equations
Stokes
Gas dynamics
equations
hyperbolic
parabolic - elliptic

37 02-Basics_CFD
 Flow Regimes

§ Different classes of PDEs

ØDifferent numerical methods

Image: game-icons.net, Lorc

38 02-Basics_CFD
 Simplification to Euler Equations

Compressible Navier-Stokes equations

include friction and thermal conduction

hyperbolic – parabolic

Re ! 1
<latexit sha1_base64="I5luM6vLngbZrl954Tr/g73LDLE=">AAAC33ichVFLS8NAEJ7GV31XxZOXYBE8lVQE9VZ84UWoYrTQlrJJt3FpXmy2lSrePYlXj171F+lv8eCXNQpaxA2bmf3mm29ndpzYF4myrNecMTI6Nj6Rn5yanpmdmy8sLJ4nUU+63HYjP5I1hyXcFyG3lVA+r8WSs8Dx+YXT3UvjF30uExGFZ2oQ82bAvFB0hMsUoFZh+ZSbDSm8S8WkjK7Mhgg7atAqFK2SpZc57JQzp0jZqkaFN2pQmyJyqUcBcQpJwfeJUYKvTmWyKAbWpBtgEp7QcU63NIXcHlgcDAa0i7+HUz1DQ5xTzURnu7jFx5bINGkN+1ArOmCnt3L4Cew79rXGvD9vuNHKaYUDWAeKk1rxGLiiSzD+ywwy5lct/2emXSnq0LbuRqC+WCNpn+63zj4iElhXR0w60EwPGo4+9/ECIayNCtJX/lIwdcdtWKYt1yphpsigJ2HT10c9GHP591CHHXujtFMqn2wWK7vZvPO0Qqu0jqFuUYWOqIoyXDTzRM/0YjjGnXFvPHxSjVyWs0Q/lvH4Aaqel4A=</latexit>
Ma ! 0
<latexit sha1_base64="bbp4zPa1w+EGLIwnaTnIEJcG+fA=">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</latexit>

Incompressible Navier-
Euler equations
Stokes
Gas dynamics
equations
hyperbolic
parabolic - elliptic

39 02-Basics_CFD
 Neglect Friction and Heat Transport

§ Compressible Navier-Stokes -> Euler equations

@⇢(~x, t)
+ r · (⇢~v ) = 0
@t <latexit sha1_base64="I/v/rYzbRvQcBiWjR59nNn3x5MQ=">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</latexit>

@ m(~
~ x, t)
+ r · (m ~ ~v + pI) = r · ⌧
@t
<latexit sha1_base64="Ls/XFY5rmRVMBqEU0IioS4xjM8U=">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</latexit>

@e(~x, t)
+ r · (e + p)~v = r · ⌧~v + r · (rT )
<latexit sha1_base64="NAVN81d/WWE6cw4xo55oS7Z/cE8=">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</latexit>
@t

40 02-Basics_CFD
 Gas Dynamics: The Euler Equations

§ Hyperbolic system (wave transport)

@⇢(~x, t)
+ r · (⇢~v ) = 0
<latexit sha1_base64="rIqKqDdB5/3K6TvRh3MndGQBJak=">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</latexit>
@t
@ m(~
~ x, t)
+ r · (m~ ~v + pI) = 0
<latexit sha1_base64="oQO8+ji2GV7U1LvvppEyUgrjk+E=">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</latexit>
@t
@e(~x, t)
+ r(e + p)~v = 0
@t
<latexit sha1_base64="5+0Kqp636VyWrt6Z1WTVQJXcXjE=">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</latexit>

+Equation of state (usually ideal gas)

Image: game-icons.net, Lorc

41 02-Basics_CFD
 Gas Dynamics

§ Nonlinear transport problem

§ State travels along characteristics
§ Formation of discontinuities (shocks)

Schlieren image
of supersonic
aircraft with
shock waves.
Image: NASA Photo

42 02-Basics_CFD
 Vectorial Notation (1D, Perfect Gas)
2 3
⇢(x, t)
§ Gather state in one vector:
~u(x, t) = 4m(x, t)5
e(x, t)
2 3
<latexit sha1_base64="F16TeNiI9lJEoBpVJDLtfVd3ahQ=">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</latexit>

m
6 (3 )m2 7
~
f (~u) = 4 ⇣2⇢ + ( 1)e 5
§ Also the flux: ⌘
m (1 )m2
⇢ e+ 2⇢
<latexit sha1_base64="fEItiQTB3iK4k7vdgzGyd/limQ8=">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</latexit>

§ (Equation of state for perfect gas in flux)

43 02-Basics_CFD
 Compact Notation of the System with Vectors

@~u(x, t) @ f~(~u)
§ First order, nonlinear PDE system + =0
<latexit sha1_base64="EEMC+rIG3g3DAYh7BkZWgcrygB0=">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</latexit>
@t @x
§ Conservative variables ~u
<latexit sha1_base64="Wwpj/O9+Fo4JNQ7fK8cxYmUTpFA=">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</latexit>
<latexit

§ Compact notation shows structure of the PDEs

44 02-Basics_CFD
 More Spatial Dimensions

§ Space coordinate becomes a vector ~x

<latexit sha1_base64="N2UDge5oZSLGseC+byq3NkA0EzA=">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</latexit>
<latexit

§ Momentum becomes a vector m

~
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<latexit

§ ( more components in the state vector)

§ Get a flux for each spatial dimension

45 02-Basics_CFD
 Compact Notation, Multiple Spatial Dimensions

§ For d dimensions we have:

d
X
@~u(~x, t) @ f~i (~u)
+ =0
@t i=1
@x i
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§ For example in 2D, 2 fluxes with 4 components:

2 3 2 3
m1 m2
6 (3 )m21 m22 7 6 m2 ·m1 7
6 2⇢ +( 1)(e ) 7
2⇢ 7~ 6 ⇢ 7
f1 (~u) = 6
~ 6 2 7
6 m1 ·m2 7f2 (~u) = 6 (3 )m22
2⇢⇣ +( 1)(e
m1
7
2⇢⌘)5
4 ⇣ ⇢ ⌘5 4
m1 (1 )(m21 +m2 )
2
m2 (1 )(m21 +m2 )
2

⇢ e + 2⇢ ⇢ e+ 2⇢
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46 02-Basics_CFD
 Simplification To Incompressible

Compressible Navier-Stokes equations

include friction and thermal conduction

hyperbolic – parabolic

Re ! 1
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sha1_base64="hVkMG36qujfrtScZMhqWemST4Is=">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</latexit>
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sha1_base64="q0AcImRoGBBX3Vvui5A4v94Kodc=">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</latexit>

Ma ! 0
<latexit sha1_base64="HqS3/c/EXuJETRyDySrCTRr3Ki8=">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</latexit>

Incompressible Navier-
Euler equations
Stokes
Gas dynamics
equations
hyperbolic
parabolic - elliptic

47 02-Basics_CFD
 Neglect Density and Temperature Changes

§ Compressible Navier-Stokes -> Incompressible

@⇢(~x, t)
+ r · (⇢~v ) = 0
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@t
@ m(~
~ x, t)
+ r · (m~ ~v + pI) = r · ⌧
@t
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@e(~x, t)
+ r · (e + p)~v = r · ⌧~v + r · (rT )
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@t
48 02-Basics_CFD
 Divergence Free Flow

§ Constant density:
§ One variable less
§ mass conservation reduces to divergence free constrained for the velocity field:

§ Note: no time dependency in this equation

r~v = 0
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49 02-Basics_CFD
 One Equation Less

§ With r~v = 0
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§ And constant temperature rT = 0

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ØThe energy balance does not provide any additional information

50 02-Basics_CFD
 Incompressible Navier-Stokes Equations

§ Parabolic – Elliptic system of PDEs:

r · ~v = 0 <latexit sha1_base64="Wkso6CdMxL2ZtT8/t9UWl+g/S2A=">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</latexit>

@~v (~x, t) p ⌧
+ r · (~v ~v + I) = r ·
§ Variables:
@t ⇢ ⇢
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Velocity
Pressure
~v
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<latexit

p
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Image: game-icons.net, Lorc

51 02-Basics_CFD
 Potential Flow

§ Incompressible and inviscid

§ Can use a scalar potential to describe the velocity field

52 02-Basics_CFD
 Numerical Methods

§ Finite Differences § Panel Method (potential flows)

§ Finite Volumes § Vortex Lattice (potential flows)
§ Finite Elements
§ continuous
§ discontinuous
§ Finite Points
§ Smoothed Particle Hydrodynamics
§ Lattice Boltzmann
§ Pseudo-Spectral
§ Boundary Elements
53 02-Basics_CFD
 Finite Differences

§ Approximates differentials by difference quotients on a grid with point values

§ Will be briefly discussed on Tuesday
§ Solver using this scheme:
§ Overture (http://www.overtureframework.org/)

54 02-Basics_CFD
 Finite Volume

§ Utilizes an integral formulation with integral means in control volumina and

the fluxes between those
§ Will be discussed on Tuesday
§ Solvers using this scheme:
§ OpenFOAM (dedicated course)
§ Code Saturne (https://www.code-saturne.org/cms/)
§ Gerris (http://gfs.sourceforge.net/wiki/index.php/Main_Page)

55 02-Basics_CFD
 Finite Elements

§ Utilizes functions in elements to represent the solution

§ Will be discussed on Wednesday (continuous) and Thursday (discontinuous)
§ Solvers using this scheme:
§ Elmer (https://www.csc.fi/web/elmer)
§ Nektar++ (https://www.nektar.info/)
§ Ateles
(http://www.apes-suite.org/pages/ateles)

56 02-Basics_CFD
 Finite Points

§ Meshfree method based on scattered point values with a solution

construction from a local point neighborhood
§ Least-Square fitting of unknowns
§ Interesting for moving/deforming boundary problems

57 02-Basics_CFD
 Smoothed-Particle Hydrodynamics

§ Meshless, lagrangian method: particles build the fluid and a kernel function
describes the “range“ of the properties of the particle
§ Especially interesting for free-surface flows
§ Solvers implementing this scheme:
§ AQUAgpusph (http://canal.etsin.upm.es/aquagpusph/)
§ Pysph (https://pysph.readthedocs.io/en/latest/)
§ FLUIDS (http://fluids3.com/)

58 02-Basics_CFD
 Lattice-Boltzmann

§ Works on the Boltzmann equation with a discrete space

§ Cellular automata on a mesoscopic level reproduce Navier-Stokes equations
in a macroscopic view
§ Solvers with this scheme:
§ Palabos (http://www.palabos.org/)
§ Musubi (http://www.apes-suite.org/pages/musubi)
§ OpenLB (http://www.openlb.net/)

59 02-Basics_CFD
 (Pseudo)-Spectral

§ Approximation of the solution by a function series

§ Highly efficient for smooth problems
§ Limitations by function choice an geometrical layout
§ Example: http://dedalus-project.org/
§ Spectral Element Method solver:
§ Nek5000 (https://nek5000.mcs.anl.gov/)

60 02-Basics_CFD
 Boundary Elements

§ Uses boundary values to define solution to integral equation

§ Requires Green‘s function to be computable for the given problem
§ Solvers for this scheme:
§ FastBEM (http://www.yijunliu.com/Software/)
§ Nemoh (https://lheea.ec-nantes.fr/logiciels-et-brevets/nemoh-presentation-
192863.kjsp)

61 02-Basics_CFD
 Panel Method

§ Represents a potential flow by superposition of various singularities

§ Singularities organized in panels to represent walls in the flow
§ Solvers implementing this scheme:
§ XFOIL (http://web.mit.edu/drela/Public/web/xfoil/)
§ Panair (http://www.pdas.com/panair.html)
§ Q-Blade (http://www.q-blade.org/)

62 02-Basics_CFD
 Vortex Lattice

§ For potential flows

§ Prandtl‘s lifting lines theory
§ Model lifting surfaces by discrete vortex lines
§ Surfaces discretized into panels with horseshoe vortices
§ Implementation:
§ OpenVOGEL (https://sites.google.com/site/gahvogel/main)

63 02-Basics_CFD
 Focus of This Course

§ We will look at the „classical“ methods (FDM, FVM and FEM)

64 02-Basics_CFD