Derivation of the basic equations of fluid flows. No particle in the fluid at this stage (next week)particle in the fluid at this stage (next week).

•Conservation of mass of the fluid.

•Conservation of mass of a solute (applies to non-sinking particles at low concentration).

•Conservation of momentum.

•Application of these basic equations to a turbulent fluid•Application of these basic equations to a turbulent fluid.

A few concepts before we get to the meat…

Tensor (Stress), Vectors (e.g. position, velocity) and scalars (e.g. T, S, CO2).

We need to define a coordinate system, and an (infinitesimal) element of volume. ( )

We assume a continuous fluid, and that all the fields of interest are differentiabledifferentiable.

The Lagrangian framework is the framework in which the laws of classical mechanics are often stated The laws of classical mechanics are often stated. The coordinates of a point describe the trajectory from

. Density, ρ, can evolve along the trajectory.By the chain rule along a parcel trajectory:

)(txv

)0(0 == txx vv

By the chain rule, along a parcel trajectory:

∂⎟⎞

⎜⎛ ∂

⎟⎞

⎜⎛ ∂xdd v ρρρρ

( ) ⎞⎛

∂∂

+∇⋅=⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=== t

utxdt

xddtd

constxconstt

vv

v

ρρρρρ

( ) ( )⎟⎠⎞

⎜⎝⎛ ∇⋅+∂∂

=→ txutdt

ttxd vvv

,),( ρρ

∇⋅+∂∂

≡

⎠⎝

utDt

D vConversion from Lagrangian to Eulerian

∂tDt

Example:Let’s assume that we are in a river that feeds on glacial melt. The water warms at a constant rate that is a function of distance from the source. If we drift down function of distance from the source. If we drift down river (A la ‘Huckleberry Fin’), the temperature increases with time (DT/Dt>0). At one point along the river, however we may see no change in temperature with time however, we may see no change in temperature with time (∂T/∂t=0), as the water arriving there is always at the same temperature. The heat flux is advective, (u∂T/∂x>0).

In short, the convective derivative is:

∇⋅+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

= utz

wy

vx

utDt

D v

∂∂∂∂∂ tzyxtDt

Mass conservation (Eulerian, differential approach): Accounting for the change in mass inside a fixed, constant-g g ,size volume:

(x-Δx/2,y+Δy/2,z+Δz/2)(x+Δx/2,y+Δy/2,z+Δz/2)

Az =ΔxΔyz

(x-Δx/2,y-Δy/2,z+Δz/2)

Az =ΔxΔyAx=

ΔzΔy

A Δ Δ

Ay=ΔxΔz

Ax=ΔzΔy

Mass=ρVx

y (x+Δx/2,y+Δy/2,z-Δz/2)Ay=ΔxΔz

Az =ΔxΔy

( ) ( ) ( ) ( )2/2/2/2/2/2/ zzzyyyxxx wwAvvAuuA

tV

Δ−Δ+Δ−Δ+Δ−Δ+−−−−−−=

∂∂ ρρρρρρρ

Mass ρV(x-Δx/2,y-Δy/2,z-Δz/2) (x+Δx/2,y-Δy/2,z-Δz/2)

( ) ( ) ( )2/2/2/2/2/2/

111zzyyxx

wwz

vvy

uuxt Δ−Δ+Δ−Δ+Δ−Δ+

−Δ

−−Δ

−−Δ

−=∂∂ ρρρρρρρ

∂∂∂∂∂ ρρ ( ) ( ) ( ) ( ) 0=⋅∇+∂∂

=∂∂

+∂∂

+∂∂

+∂∂ u

tw

zv

yu

xtvρρρρρρ

Mass conservation (Eulerian, integral approach): Accounting for the change in mass inside a fixed Accounting for the change in mass inside a fixed, constant-volume volume (V0):

⋅−= ∫∫∂

dSnudVdtd

VV

vvρρ00

⋅∇−=∂∂

→ ∫∫

∂

dVudV

dt VV

vρρ

0

00

∇∂

∂ ∫∫ t VV

v

ρρ

0=⋅∇+∂

→ ut

ρρ

Where we used the divergence theorem:g

It states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary (WIKI).

Reiteration (no sinks/sources):

Mass conservation (Lagrangian, integral):

0=∫ dVDtD ρDt V

(E l l) Mass conservation (Eulerian, integral):

d dSnudVdtd

VV

vv ⋅−= ∫∫∂ 00

ρρ

Mass conservation: Note that:

( ) 0=⋅∇+∂∂ u

trρρ

Can be written as:

∂t

01=⋅∇+ u

DD rρ

The 2nd term is the fluid divergence (rate of outflow of volume per unit

Dtρ

The 2 term is the fluid divergence (rate of outflow of volume per unit volume). This can be nonzero only for compressible fluids. It is the rate of loss of density due to compression/expansion.

F b th t d i th t i t f th i 0∇rFor both water and air we can assume that in terms of their

dynamics (we need compressibility to pass sound…).0=⋅∇ ur

Mass balance for conserved scalar: Adding molecular diffusion:ng mo cu ar ffus on

( )∫∫ ⋅∇−−=∂∂ dSnCKuCdV

tC rr

∫∫ ∂ SV t

Where V is the volume of the control volume and S its surface, and using Fick’s law. By the help of the divergence theorem:

⎤⎡∂C ( ) 0=⎥⎦⎤

⎢⎣⎡ ∇−⋅∇+∂∂

∫ dVCKuCtC

V

r

Since the volume is arbitrary, this can be true if and only if:

( ) ( )C∂ ( ) ( )CKuCtC

∇⋅∇=⋅∇+∂∂ r

Momentum balance (Navier-Stokes): Newton’s 2nd law of motion states that the time rate of change of gmomentum of a particle is equal to the force acting on it. This law is Lagrangian, the “time rate of change” is with respect to a reference system following the particle.

dSTdVgdVudtd

tVtVtV∫∫∫

∂

+=)()()(

vvv ρρ

Where g is the body force per unit mass (e.g. gravity) and T is the surface force per unit surface area bounding V. f h l ll h h d b k f h l

dt tVtVtV ∂ )()()(

If the volume is small enough, the integrands can be taken out of the integral:

( )VddVddVd δvvv ⎟⎞

⎜⎛

∫∫

( ) dVddd vv δ

( )Vudt

dVudt

dVudt tVtV

δρρρ vvv =⎟⎟⎠

⎜⎜⎝

= ∫∫)()(

( ) ( )dtudV

dtVdu

dtudVVu

dtd vv ρδρδρδδρ =+=

Momentum balance (Navier-Stokes):

Th b d f i i il l t t d

VgdVg δρρ vv =∫

The body force is similarly treated:

ggtV

ρρ∫)(

Defining a stress tensor (expanded on the next slide):Defining a stress tensor (expanded on the next slide):

nT vv⋅Τ=

A d l i th di thAnd applying the divergence theorem:

VdVdST δΤ⋅∇=Τ⋅∇= ∫∫v

tVtV∫∫

∂ )()(

uDvΤ⋅∇+=→ g

DtuD vρρ

Surface forcing:For an inviscid fluid, the surface force exerted by the surrounding , y gfluid is normal to the surface, i.e. , and p is called the pressure force.

npT vv⋅−=

In general, viscous stress force, S, is also present. For viscous fluids: . By definition , and we now have

, where and I is the identity tensor.∑+−= IT pSnpTvvv

+⋅−= nT vv⋅Τ=

nS vv⋅∑=

For Newtonian fluids,

uv2∇=Σ⋅∇ μ

And the resultant Navier-Stokes equations for incompressible fluids are:

upgDt

uD vvv

2∇+∇−= μρρ

And the resultant Navier Stokes equations for incompressible fluids are:

Dt

Rotational symmetry:

⎟⎟⎟⎞

⎜⎜⎜⎛

=⎟⎟⎟⎞

⎜⎜⎜⎛

= yzyyxy

xzxyxx

yzyyyx

xzxyxx

ττττττ

ττττττ

T⎟⎠

⎜⎝

⎟⎠

⎜⎝ zzyzxzzzzyzx ττττττ

( )1 ( )zzyyxxp τττ ++≡−31

⎞⎛ ⎞⎛ ∂∂⎞⎛ ∂∂∂

Total stress tensor, Newtonian fluid:

⎟⎞

⎜⎛ ∂

+∂

+⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

⎟⎟⎞

⎜⎜⎛ ∂

+∂∂

+⎟⎟⎞

⎜⎜⎛ ∂

+∂

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+−

ij uuwvvvuxw

zu

xv

yu

xup

δ

μμμ

2

2

T

ij∑

⎟⎟⎠

⎜⎜⎝ ∂

+∂

+−=

⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜

∂∂

+−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎟⎠

⎜⎜⎝ ∂

+∂∂

+−⎟⎟⎠

⎜⎜⎝ ∂

+∂

=j

i

i

jij xx

p

wpwvwuyzy

pxy

μδ

μμμ

μμμ

2

2T

i, j={1, 2, 3}={x, y, z}⎟⎠

⎜⎝ ∂⎟

⎠⎜⎝ ∂∂⎠⎝ ∂∂ z

pyzxz

μμμ

Stokes, 1845: 1 Σ linear function of velocity gradients

, j { , , } { , y, }

222

1. Σij linear function of velocity gradients.2. Σij should vanish if there is no deformation of fluid elements.3. Relationship between stress and shear should be isotropic.

( )

( )xy

vyu

xu

xp

zT

yT

xTi zxyxxx

22

2

2

2

2

2

2Tˆ∂∂

∂+

∂∂

+∂∂

+∂∂

−=∂∂

+∂

∂+

∂∂

=⋅∇⋅ μμμ

( ) uxp

xuu

xp

xzw

zu 22

2

2

2

∇+∂∂

−=∂⋅∇∂

+∇+∂∂

−=∂∂

∂+

∂∂

+ μμμμv

Navier-Stokes equations:

0u ,

222

2

⎞⎛

=⋅∇+∇+∇

−= gupDt

uDρμ

ρvvv

v

1

222

2

2

2

2

2

2

⎞⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

zu

yu

xu

xp

zuw

yuv

xuu

tu

ρμ

ρ

12

2

2

2

2

2

⎞⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

zv

yv

xv

yp

zvw

yvv

xvu

tv

ρμ

ρ

12

2

2

2

2

2

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂ g

zw

yw

xw

zp

zww

ywv

xwu

tw

ρμ

ρ

0=∂∂

+∂∂

+∂∂

zw

yv

xu

Coriolis is added when moving the framework to an accelerating framework. Have to add boundary & initial conditions.

Navier-Stokes equations (Boussinessq approximation):Separate balance of fluid at rest from moving fluid.

( ) ( ),,,'0 += tzyxpzpp( ),,,'0 += ρρρ tzyx

First order balance (hydrostatic):

10 0 +∂

−= gpFirst order balance (hydrostatic):

00

+∂

=ρ

gz

2 d d b l :

0u,' 02

0 =⋅∇+∇+−∇= vvv

ρμρ gupuD2nd order balance:

0u ,00 ∇+∇+∇ ρμρ gupDt

Example: steady flow under gravity down an inclined plane.

z

g

α

∂∂ From: Acheson 1990

0zw

xu

=∂∂

+∂∂ From: Acheson, 1990

ανρ

sin10 2

2

gzu

xp

+∂∂

+∂∂

−=

α

ρ

cos10

0

gpzx

+∂

−=

∂∂BCs: uwz

∂=== 0 :0

αρ

cos00

gz+

∂=

appzuhz ==∂∂

= ,0 : μ

Example: steady flow under gravity down an inclined plane.

z

g

α

( )Solution:

( )

( )

αρ cos0

gzhgpp a −=−

( ) αν

sin22

zhzgu −=

Q: what ν should we use?

Reynolds decomposition of the N-S equations

Assume a turbulent flow. At any given point in space we separate the mean flow (mean can be in time, space, or ensemble) and deviation from the mean such that:

',',' ,' wwuvvuuuuppp +=+=+=+=ensemble) and deviation from the mean such that:

',0' , etcppp ==

Substituting into the continuity equation (linear):

0''' ;0 =∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

zw

yv

xu

zw

yv

xu

yy

⎟⎞

⎜⎛ ∂∂∂∂∂∂∂∂ uuupuuuu 1 222μ

Substituting into x-momentum Navier-Stokes equation:

( ) ( ) ( )⎞⎛ ∂∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

zu

yu

xu

xp

zuw

yuv

xuu

tu

''''''

1222ρ

μρ

( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

+∂

∂−

zuw

yuv

xuu ''''''

The evolution of the mean is forced by correlations of fluctuating properties.The correlation terms time density are the “Reynolds stresses”.

Substituting into a scalar conservation equation:Substituting into a scalar conservation equation

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

zC

yC

xCK

zCw

yCv

xCu

tC

2

2

2

2

2

2

ρ

( ) ( ) ( )⎟⎟⎞

⎜⎜⎛

∂∂

+∂

∂+

∂∂

−

⎠⎝ ∂∂∂∂∂∂∂

CwCvCu

zyxzyxt

''''''

ρ

⎟⎠

⎜⎝ ∂∂∂ zyx

Note that the Reynolds stress tensor is symmetric (as is the viscous stress tensor):the viscous stress tensor):

⎟⎞

⎜⎛ '''''' wuvuuu

⎟⎟⎟⎟⎞

⎜⎜⎜⎜⎛

−= ''''''0 wvvvvuwuvuuu

ρτ⎟⎟⎠

⎜⎜⎝ '''''' wwwvwu

The closure problem: to develop equations for the evolution of the Reynolds stresses theselves, higher order correlation are needed (e.g. w’u’u’) and so on. For this reason theories have been devised to describe τij in terms of th flthe mean flow.

For more, see: http://www.cfd-online.com/Wiki/Introduction to turbulence/Reynolds averaged equationsonl ne.com/W k /Introduct on_to_turbulence/Reynolds_averaged_equat ons

One solution to the closure problem is to link the Reynolds’ stress to mean-flow Quantities. For example:

UKuw dd∂

=− ''0ρ

TKTw

zKuw

eddy

eddy

∂∂

=−

∂

''0

0

ρ

ρ

This type of formulation is appealing because it:a. Provide for down-gradient flux.b f l l d ff d

zeddy ∂0ρ

b. Is reminiscent of molecular diffusion and viscosity.c. Provide closure to the equations of the the mean fields.

This type of formulation is problematic because:This type of formulation is problematic because:a. Keddy is a property of the flow and not the fluid. b. Keddy is likely to vary with orientation, unlike molecular processes.

How is Keddy related to the turbulence?

Assume a gradient in a mean property (momentum, heat, solute, etc’. Remember: no mean gradient no flux). Assume a fluctuating velocity field:

l’ is the distance a parcel travels before it loses its identity. The rate of upward vertical turbulent transfer of <Ψ> is down the mean gradient:

Klwlw eddy ∂∂

−=∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

+ψψψψ ''''zzz eddy ∂∂⎠⎝ ∂

ψ

How is Keddy related to the turbulence?

TzUKuw eddy

∂∂∂

=− ρρ 00 ''

T k d L l (1972) h thi bl f di i l l i

zTTw T ∂∂

=− γρρ 00 ''

Tennekes and Lumley (1972) approach this problem from dimensional analysis based on assuming a single length scale-l and a single velocity scale ω=<w’2>1/2.

2 )1(~ ;'' 200 Occuw ωρρ =−

The eddies involved in momentum transfer have vorticities ω/l; this The eddies involved in momentum transfer have vorticities, ω/l; this vorticity is maintained by the mean shear (l is the length scales of the eddies, e.g. the decorrelation scale).

U∂ )1(~ ;/ OczUcl∂∂

=ω

It follows that:

UllK dd∂2~~ ω

zllK ddye ∂

ω

In analogy with momentum flux, for heat we have:

zTcTwc Tpp ∂∂

=− γρρ 00 ''

It is most commonly assumed, and verified that γT=Keddy.

Eddy-diffusion: perspective from a dye patch (figures from lecture notes of Bill Young, UCSD)

Dye patch << dominant scale of eddies. Dashed circle denotes initial position of patch.

Dye patch ≈ dominant scale of eddiesDye patch dominant scale of eddies

Dye patch >> dominant scale of eddies

Cheat sheet:

1 G di f l ( )

kji ∂+

∂+

∂=∇

φφφφ ˆˆˆ

1. Gradient of a scalar (a vector):

zk

yj

xi

∂+

∂+

∂∇φ

∂∂∂ ϕ2. Divergence of a vector (a scalar):

zyxzyx

∂∂

+∂

∂+

∂∂

=⋅∇ϕϕϕϕv

3 Divergence of a Tensor T =T :

+⎟⎟⎞

⎜⎜⎛ ∂

+∂

+∂

+⎟⎟⎞

⎜⎜⎛ ∂

+∂

+∂

=⋅∇TTT

jTTTi zyyyxyzxyxxx ˆˆT

3. Divergence of a Tensor, T12=T(face)(direction):

⎟⎟⎞

⎜⎜⎛ ∂

+∂

+∂

⎟⎟⎠

⎜⎜⎝ ∂∂∂⎟⎟

⎠⎜⎜⎝ ∂∂∂

TTTk

zyxj

zyx

zzyzxzˆ⎟⎟⎠

⎜⎜⎝ ∂

+∂

+∂ zyx

k

Cheat sheet (cotinued):

4 L l i f ( )4. Laplacian of a vector (a vector):

( ) +⎟⎟⎞

⎜⎜⎛

∂∂

+∂∂

+∂∂

=⋅∇⋅∇=∇ 2

2

2

2

2

22 i xxx ϕϕϕϕϕ vv ( )

⎟⎟⎞

⎜⎜⎛ ∂

+∂

+∂

+⎟⎟⎞

⎜⎜⎛ ∂

+∂

+∂

⎟⎠

⎜⎝ ∂∂∂

222222

222

ˆˆ kj

zyx

zzzyyy ϕϕϕϕϕϕ

ϕϕ

⎟⎟⎠

⎜⎜⎝ ∂

+∂

+∂

+⎟⎟⎠

⎜⎜⎝ ∂

+∂

+∂ 222222 zyx

kzyx

j