Fluidmechanik turbulenter Stromungen Markus Uhlmann - KIT fileEquations of uid motion Transformation...

$: Fluidmechanik turbulenter Stromungen Markus Uhlmann - KIT fileEquations of uid motion Transformation properties Film \Turbulence" Fluid mechanics of turbulent ows Fluidmechanik turbulenter$
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

http://www.ifh.kit.edu/213_216.php



LECTURE 2

Equations of fluid motion

2/16



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



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

4/16




Mass conservation: continuity equation

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp



ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp

pppppppppppppppp

ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp

pppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

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

5/16




Newton’s 2nd law: momentum equation

Momentum balance

I ρDuDt︸︷︷︸

material acceleration

= ρ f︸︷︷︸body forces

+ P︸︷︷︸surface forces

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppp

pppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppp.............................................................................

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

6/16




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

7/16




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




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)

9/16




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




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




Vortex stretching and tilting

Contributions to the term (ω · ∇)u





pppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ωyy

x

v

I ωy ∂y(v) > 0

→ tends to increase ωy

I “vortex stretching”


pppppppppppppppppppp pppppppppppppppppppp

pppppppppppppppppppp pppppppppppppppppppp ppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp

ωyy

x

u

I ωy ∂y(u) > 0

→ tends to generate ωx

I “vortex tilting”

12/16




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



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



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)

15/16



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

http://web.mit.edu/fluids/www/Shapiro/ncfmf.html

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


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


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

http://web.mit.edu/fluids/www/Shapiro/ncfmf.html


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


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


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


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


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


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


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


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


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


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


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

Fluidmechanik turbulenter Stromungen Markus Uhlmann - KIT fileEquations of uid motion Transformation...

Documents

Transcript of Fluidmechanik turbulenter Stromungen Markus Uhlmann - KIT fileEquations of uid motion Transformation...