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Equations of fluid motionTransformation properties
Film “Turbulence”
Fluid mechanics of turbulent flowsFluidmechanik turbulenter Stromungen
Markus Uhlmann
Institut fur HydromechanikKarlsruher Institut fur Technologie
www.ifh.kit.edu
WS 2010/2011
Lecture 2
1/16
Equations of fluid motionTransformation properties
Film “Turbulence”
LECTURE 2
Equations of fluid motion
2/16
Equations of fluid motionTransformation properties
Film “Turbulence”
Questions to be answered in the present lecture
How can the fluid motion be described mathematically?
What are the transformation properties of theconservation laws?
Visual Impressions? Film “Turbulence” by R.W. Stewart
3/16
Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
General assumptions
Set the following framework:
I continuum hypothesis (Kn ≡ λ/`� 1)
I consider Newtonian fluids
I restrict to incompressible fluids
Invoke the following principles:
I mass conservation
I Newton’s second law
+ Material properties
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Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
Mass conservation: continuity equation
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ρu+ ∂ρu∂x
dxρu
x
z
y
ρv
ρv + ∂ρv∂y
dy
ρw + ∂ρw∂z
dz
ρw
I time: t
I space: x = (x, y, z)
I density: ρ(x, t)
I velocity:u(x, t) = (u, v, w)
I∂ρ
∂t+∂(ρ ui)∂xi
= 0 incompressibility:∂ui∂xi
= 0
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Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
Newton’s 2nd law: momentum equation
Momentum balance
I ρDuDt︸ ︷︷ ︸
material acceleration
= ρ f︸︷︷︸body forces
+ P︸︷︷︸surface forces
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x
z
y σxx + ∂σxx
∂xdx
σxy + ∂σxy
∂xdx
σxz + ∂σxz
∂xdx
σxx
σxzσxy
IDuDt
=∂u∂t
+(u·∇) u
I P=−∇p+∇ · τ
I Newtonian fluid:τij = µ(ui,j + uj,i)
I gravity: f = g
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Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
Momentum equation
Navier-Stokes equation for incompressible Newtonian fluid:
I ∂tu + (u · ∇) u +1ρ∇p = ν∇2u + f
I alternatively, in tensor notation:
∂tui + ujui,j +1ρp,i = ν ui,jj + fi
I constant density: gravity absorbed in modified pressure:
∂tu + (u · ∇) u +1ρ∇p = ν∇2u
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Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
The role of pressure
Consider constant-density flows
I pressure 6= thermodynamic variable
I pressure enforces divergence-free condition (∆ ≡ ∇ · u):
D∆Dt− ν∇2∆ = −1
ρ∇2 p− uj,i ui,j︸ ︷︷ ︸
≡R
for ∆ = 0, need R = 0. Therefore: ∇2p = −ρ(uj,i ui,j)
I non-local character of pressure (global integral over x′):
p(x, t) = ph(x, t) +ρ
4π
∫uj,i ui,j|x− x′|
dx′
⇒ perturbations affect the entire flow
8/16
Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
The kinetic energy equation
Derivation
I definition of kinetic energy:Ek ≡ u · u/2 = ui ui/2 (summation on i)
I transport equation: dot product of u with momentum equ.
DEkDt
= − (uj p/ρ),j + (2νuiSij),j − 2νSijSij
with: Sij = 12(ui,j + uj,i)
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Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
The kinetic energy equation
Significance of the terms
I∂Ek∂t
= − (Ekuj),j −(p
ρuj
),j
+ (2νuiSij),j − 2νSijSij
I integration over an arbitrary volume V:
d
dt
∫EkdV = −
∮(Eku · n) dS︸ ︷︷ ︸convection
−∮ (
p
ρu · n
)dS︸ ︷︷ ︸
pressurework
+∮
(u · τ ) · n dS︸ ︷︷ ︸viscouswork
−∫
(2νSijSij) dV︸ ︷︷ ︸dissipation
I rate of dissipation: ε ≡ 2νSijSij ≥ 0
10/16
Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
The vorticity equation
Definition of vorticity
I ω = ∇× u =(∂w∂y −
∂v∂z ,
∂u∂z−
∂w∂x ,
∂v∂x−
∂u∂y
)TI (twice) angular velocity of fluid element
I derive transport equation by applying curl to momentumequation:
∂tω + (u · ∇)ω = (ω · ∇) u + ν∇2ω
I pressure eliminated, additional stretching term
11/16
Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
Vortex stretching and tilting
Contributions to the term (ω · ∇)u
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ωyy
x
v
I ωy ∂y(v) > 0
→ tends to increase ωy
I “vortex stretching”
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ωyy
x
u
I ωy ∂y(u) > 0
→ tends to generate ωx
I “vortex tilting”
12/16
Equations of fluid motionTransformation properties
Film “Turbulence”
ContinuityMomentumKinetic energyVorticity
The enstrophy equation
Enstrophy and vortex stretching
I definition of enstrophy: ω2/2 = ω · ω/2I transport equation: multiply vorticity equation by ω
D(ω2/2)Dt
= ω · (ω · ∇) u + ν∇2(ω2/2)− ν(∇ω)2
I in a turbulent flow:the stretching term tends to increase the enstrophy
13/16
Equations of fluid motionTransformation properties
Film “Turbulence”
Non-dimensional equations
I use length scale L and velocity scale UI define non-dimensional variables:
x = x/L, t = t/T = tU/Lu = u/U , p = p/(ρU2)
I substitute into Navier-Stokes equations:
∂ui∂xi
= 0
∂ui
∂t+ uj
∂ui∂xj
= − ∂p
∂xi+
1Re
∂2ui∂xj∂xj
I Reynolds number: Re ≡ ULν
14/16
Equations of fluid motionTransformation properties
Film “Turbulence”
Transformation properties
Navier Stokes equations
I Reynolds number similarity(different Lb, Ub, νb only changes Reb)
I time and space invariance(shift by X and T )
I rotated or reflected reference frame → invariant
I time reversal: NOT invariant(viscous term changes sign!)
I Galilean invariance (constantly moving frame)→ Navier-Stokes are invariant (velocities are not)
I frame rotating in time → not invariant (Coriolis force)
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Equations of fluid motionTransformation properties
Film “Turbulence”
Film material
“Turbulence”
I by R.W. Stewart, University of British Columbia
I 29 min
I US Natl. Commitee on Fluid Mechanics Films (1969)
I Weblink I
16/16
ConclusionOutlookFurther readingNotation
Summary
Main questions of the present lecture
I How can the fluid motion be described mathematically?I Navier-Stokes equations (momentum + continuity)
I alternatively: vorticity equation
⇒ energy, enstrophy equations
I What are the transformation properties of theconservation laws?
I Re similarity, rotation/reflection, Galilean invariance
I NO time reversal, Coriolis in rotating frame
1/14
ConclusionOutlookFurther readingNotation
Outlook on next lecture: Statistical description
How do we compute and analyze a turbulent flow statistically?
Part I: What are the basic mathematical tools?
Part II: What are the averaged equations?
2/14
ConclusionOutlookFurther readingNotation
Further reading
I S. Pope, Turbulent flows, 2000→ chapter 2
I U. Frisch, Turbulence, 1995→ chapter 2
I US Natl. Commitee on Fluid Mechanics Films (Weblink)
3/14
ConclusionOutlookFurther readingNotation
Notation
P
3
2
1
x
x
x
O
x
ee
e
2
1
3
3
12
Vectors
I Cartesian coordinate system with unit vectors: e1, e2, e3
→ position vector: x = x1e1 + x2e2 + x3e3 = xe1 + ye2 + ze3
I denote vector by: x =
xyz
note: boldface
and its transpose: xT = (x, y, z)4/14
ConclusionOutlookFurther readingNotation
Notation (2)
3
2
1
1’
3’
2x
O
Rotated coordinate system
I components of vector x in rotated coordinate system:
x′j = xiCij (1)
with rotation matrix components Cij (Cij = cosαij in 2D)
→ a vector is a quantity that transforms as in (1)
5/14
ConclusionOutlookFurther readingNotation
Notation (3)
Second order tensor
I extension of a vector, with two directional indices
I a matrix is a tensor if it satisfies the transformation:
τ ′mn = CTmi τij Cjn (2)
I note: summation convention over repeated indices
CTmi τij Cjn implies:∑3
i=1
∑3j=1C
Tmi τij Cjn
I notation for tensors: τ , A, etc.
6/14
ConclusionOutlookFurther readingNotation
Notation (4)
Example: stress tensor
I τ =
τ11 τ12 τ13
τ21 τ22 τ13
τ31 τ32 τ33
I first index: orientation of the surface
I second index: direction of the force on that surface
7/14
ConclusionOutlookFurther readingNotation
Notation (5)
Kronecker delta
I defined as: δij ={
1 if i = j0 if i 6= j
I in matrix form: δ = I =
1 0 00 1 00 0 1
8/14
ConclusionOutlookFurther readingNotation
Notation (6)
The alternating symbol
I also called: permutation tensor (3rd order tensor)
I defined as:
εij =
1 if ijk = 123, 231, or 312 (cyclic order)0 if any two indices are equal−1 if ijk = 321, 213, or 132 (anticyclic order)
I used in definition of curl
9/14
ConclusionOutlookFurther readingNotation
Notation (7)
Dot product (inner product)
I dot product between two vectors u and v is a scalar:
u · v = v · u = uivi
Cross product
I cross product between vectors u and v is a vectorperpendicular to plane of u, v:
u×v = (u2v3−u3v2)e1 + (u3v1−u1v3)e2 + (u1v2−u2v1)e3
I index notation: (u× v)k = εijk ui vj
10/14
ConclusionOutlookFurther readingNotation
Notation (8)
Gradient and nabla operator
I definition: ∇ ≡ e1∂∂x + e2
∂∂y + e3
∂∂z = ei ∂
∂xi
I operating on a scalar generates vector: ∇φ = ei∂φ∂xi
I the gradient of φ in a direction n is given by: ∂φ∂n = (∇φ) · n
Divergence
I divergence of a vector is a scalar: ∇ · u = ∂ui∂xi
I divergence of a 2-tensor is a vector: (∇ · τ )i = ∂τij∂xj
11/14
ConclusionOutlookFurther readingNotation
Notation (9)
Curl of a vector
I defined as cross product between nabla operator and a vector:
(∇× u)i = εijk∂uk∂xj
I components are: ∇× u =
∂w∂y −
∂v∂z
∂u∂z −
∂w∂x
∂v∂x −
∂u∂y
I can be remembered from determinant:
∇× u =
∣∣∣∣∣∣e1 e2 e3
∂x ∂y ∂zu v w
∣∣∣∣∣∣12/14
ConclusionOutlookFurther readingNotation
Notation (10)
Some useful identities
I ∇× (∇φ) = 0 (for any scalar φ)
I ∇ · (∇× a) = 0 (for any vector a)
I dyadic (tensor) product between two vectors:
A = u⊗ vyields a 2nd order tensor with components:
Aij = uivj
13/14
ConclusionOutlookFurther readingNotation
Notation (11)
Shorthand notation
I abbreviation of partial derivative expressions:
∂t ≡∂
∂t, ∂x ≡
∂
∂x
I the following is often used equivalently:
u,t ≡∂u
∂t, u,x ≡
∂u
∂x, ui,j ≡
∂ui∂xj
I example: ∇ · u = ui,i =3∑i=1
∂ui∂xi
=∂u
∂x+∂u
∂y+∂u
∂z
14/14