conservation mass - university of notre damegtryggva/cfd-course2017/lecture-4...the conservation law...

Computational Fluid Dynamics


http://www.nd.edu/~gtryggva/CFD-Course/

Grétar Tryggvason

Lecture 4January 30, 2017


The Equations Governing Fluid

Motion

http://www.nd.edu/~gtryggva/CFD-Course/


Derivation of the equations governing fluid flow in integral form

Conservation of MassConservation of MomentumConservation of Energy

Differential formSummaryIncompressible flowsInviscid compressible flows

OutlineComputational Fluid Dynamics

Conservationof

Mass


In general, mass can be added or removed:

time tThe conservation law must be stated as:

OriginalMass

MassAdded

FinalMass += Mass

Removed-

time t+∆t

Conservation of MassComputational Fluid Dynamics

We now apply this statement to an arbitrary control volume in an arbitrary flow field

Conservation of Mass

Rate of increase of mass

Net influx of mass=


Control volume VControl surface S

∂∂t

ρdvV∫

The rate of change of total mass in the control volume is given by:

Total mass

Rate of change

Conservation of MassComputational Fluid Dynamics

Control volume VControl surface S

To find the mass flux throughthe control surface, let�s examinea small part of the surface where the velocity is normal to the surface



U∆t

∆A

Mass flow through the boundary ∆A during ∆t is: time t time t+∆t

Density ρVelocity Ux

Select a small rectangle outside the boundary such that during time ∆t it flows into the CV:


where Ux = -u·n

ρUxΔtΔA


The sign of the normal component of the velocity determines whether the fluid flows in or out of the control volume. We will take the outflow to be positive:

− ρu ⋅ndsS∫

The net in-flow through the boundary of the control volume is therefore:

Mass flow normal to boundary

Integral over the boundary

Negative since this is net inflow

�

u ⋅n < 0 Inflow



∂∂t

ρdvV∫ = − ρu ⋅nds

S∫

Conservation of Mass Equation in Integral Form


Net inflow of massRate of change of mass


Conservationof

momentum


Rate of increase of momentum

Net influx of momentum

Body forces

Surface forces

= + +

Momentum per volume:

Momentum in control volume:

ρu

ρu dvV∫

Conservation of momentumComputational Fluid Dynamics

Control volume VControl surface SThe rate of

change of momentum in the control volume is given by:

Total momentum

Rate of change

∂∂t

ρu dvV∫

Rate of change of momentum


Ux∆t

∆A

Momentum flow through the boundary ∆A during ∆t is: time t time t+∆t

Density ρVelocity u

Select a small rectangle outside the boundary such that during time ∆t it flows into the CV:

− ρuu ⋅nds∫

In/out flow of momentum

Ux

Ux = -u·n so the net influx of momentum per unit time (divided by ∆t ) is:

ρu( )UxΔtΔA


Control volume V

Body forces, such as gravity act on the fluid in the control volume.

Total body force

ρ f dvV∫

f

dv

Body force


Control volume VControl surface SThe viscous

force is given by the dot product of the normal with the stress tensor T

Total stress force

T ⋅n

S∫ dS

T•nds

Viscous forceComputational Fluid DynamicsViscous flux of momentum

�

T = −p + λ∇⋅u( )I + 2µD

�

D =12∇u +∇uT( )

�

Dij =12

∂ui∂x j

+∂uj∂xi

⎛

⎝ ⎜ ⎞

⎠ ⎟

Stress Tensor

Deformation Tensor

Whose components are:

�

∇ ⋅u = 0

�

T = −pI+ 2µD

For incompressible flows

And the stress tensor is


Control volume VControl surface SThe pressure

produces a normal force on the control surface. The total force is:

Total pressure force

pn

S∫ dS

pn

ds

Pressure forceComputational Fluid DynamicsConservation of Momentum

∂∂t

ρudv∫ = − ρuu ⋅nds − Tn∫ ds∫ + ρ f dv∫

∂∂t

ρudv∫ = − ρuu ⋅nds − pn∫ ds∫ + 2µD∫ ⋅nds + ρ f dv∫

Gathering the terms

Substitute for the stress tensor

Rate of change of momentum

Net inflow of momentum Total

pressureTotal viscous force

Total body force

�

T = −pI+ 2µD


Conservationof

energy


∂∂t

ρ e + 12u2⎛

⎝⎜⎞⎠⎟dv∫ = − ρ e + 1

2u2⎛

⎝⎜⎞⎠⎟u ⋅nds∫

+ u ⋅ f dv +V∫ n ⋅ (uT)ds∫ − n ⋅qds∫

Conservation of energy

Rate of change of kinetic+internal energy

Work done by body forces

Net inflow of kinetic+internal energy

Net heat flow

Net work done by the stress tensor

The energy equation in integral form


Differential Formof

the Governing Equations


The Divergence or Gauss Theorem can be used to convert surface integrals to volume integrals

∇ ⋅aV∫ dv = a ⋅n ds

S∫

Differential form


Start with the integral form of the mass conservation equation∂∂t


S∫Using Gauss�s theorem

ρu ⋅nds =S∫ ∇ ⋅(ρu)dv

V∫The mass conservation equations becomes

∂∂t

ρdvV∫ = − ∇ ⋅(ρu)dv

V∫

Differential formComputational Fluid Dynamics

Rearranging∂∂t

ρdvV∫ + ∇⋅ (ρu)dv

V∫ = 0

∂ρ∂t

+ ∇⋅ (ρu)⎛ ⎝

⎞ ⎠ dvV∫ = 0

Since the control volume is fixed, the derivative can be moved under the integral sign

Rearranging

∂ρ∂tdv

V∫ + ∇⋅(ρu)dvV∫ = 0

Differential form


∂ρ∂t

+∇ ⋅(ρu) = 0

∂ρ∂t

+ ∇⋅ (ρu)⎛ ⎝

⎞ ⎠ dvV∫ = 0

This equation must hold for ANY control volume, no matte what shape and size. Therefore the integrand must be equal to zero


�

DρDt

+ ρ∇ ⋅u = 0

�

∇ ⋅ (ρu) = u ⋅ ∇ρ + ρ∇ ⋅u

The mass conservation equation becomes:

Expanding the divergence term:

�

∂ρ∂t

+ ∇ ⋅ (ρu) = ∂ρ∂t

+ u ⋅ ∇ρ + ρ∇ ⋅u

or

where

�

D()Dt

= ∂()∂t

+ u ⋅ ∇() Convective derivative

Differential form

Computational Fluid DynamicsDifferential form

�

∂∂t

ρu dv∫ = ρ f dv∫ + (nT− ρu(u ⋅n))ds∫

The differential form of the momentum equation is derived in the same way:Start with

Rewrite as:

�

∂∂t

ρu dv∫ = ρ f dv∫ + ∇ ⋅∫ T− ρuu( )dv

To get:∂ρu∂t

= ρ f +∇⋅(T − ρuu)


Using the mass conservation equation the advection part can be rewritten:

�

∂ρu∂t

+ ∇ ⋅ ρuu =

u ∂ρ∂t

+ u∇ ⋅uρ + ρ ∂u∂t

+ ρu ⋅ ∇u =

u ∂ρ∂t

+ ∇ ⋅uρ⎛ ⎝ ⎜

⎞ ⎠ ⎟ + ρ Du

Dt= ρ Du

Dt

Differential form

=0, by mass conservation


The momentum equation equation

∂ρu∂t

= ρ f +∇ ⋅ (T − ρuu)

�

ρDuDt

= ρ f + ∇⋅T

Can therefore be rewritten as

Differential form

where

�

ρ DuDt

= ρ ∂u∂t

+ u ⋅ ∇u⎛ ⎝ ⎜

⎞ ⎠ ⎟


The energy equation equation can be converted to a differential form in the same way. It is usually simplified by subtracting the �mechanical energy��

Differential form


The �mechanical energy equation� is obtained by taking the dot product of the momentum equation and the velocity:

�

ρ∂∂t

u2

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ = −ρu ⋅ ∇ u2

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ + ρu ⋅ fb + u ⋅ ∇ ⋅T( )

The result is:

The �mechanical energy equation�

�

u ⋅ ρ ∂u∂t

+ ρu ⋅ ∇u = ρ f + ∇⋅T⎛ ⎝

⎞ ⎠


The final result is

�

ρDeDt

− T ⋅ ∇u +∇ ⋅q = 0

Subtract the �mechanical energy equation� from the energy equation

Using the constitutive relation presented earlier for the stress tensor and Fourie�s law for the heat conduction:

�

q = −k∇T

�

T = −p + λ∇⋅u( )I + 2µD

Differential form


The final result is

�

ρDeDt

+ p∇⋅u =Φ+∇ ⋅ k∇T

is the dissipation function and is the rate at which work is converted into heat.

�

µ, λ, k

�

p = p(e,ρ); T = T (e,ρ);

where

�

Φ = λ(∇⋅u)2 + 2µD ⋅D

Generally we also need:

and equations for


Summaryof

governing equations


∂∂t


S∫

∂∂t

ρudv∫ = ρ f dv∫ + (nT − ρu(u ⋅n))ds∫

∂∂t

ρ(e + 12u2 )dv∫ = u ⋅ ρ f dv∫ + n ⋅ (uT − ρ(e + 1

2u2 ) − q)ds∫

Integral Form


�

∂ρ∂t

+ ∇ ⋅ (ρu) = 0

�

∂ρu∂t

= ρ f + ∇ ⋅ (T− ρuu)

�

∂∂t

ρ(e + 12u2)=

∇ ⋅ ρ(e + 12u2)u −uT+q

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Conservative Form

DρDt

+ ρ∇⋅u = 0

ρDuDt

= ρ f +∇ ⋅T

ρDeDt

= T∇⋅u − ∇⋅q

Convective Form


Special CasesCompressible inviscid flowsIncompressible flows

Stokes flowPotential flows Not in this course!


Inviscid compressible flows


∂U∂t

+∂E∂x

+∂F∂y

+∂G∂z

= 0

U =

ρρu

ρv

ρw

ρE

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

E =

ρuρu2 + p

ρuv

ρuw

ρE + p( )u

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

F =

ρvρuv

ρv2 + p

ρvw

ρE + p( )v

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

G =

ρuρuw

ρvw

ρw2 + p

ρE + p( )u

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

�

p = ρRT; e = cvT; h = c pT; γ = c p /cv

�

cv =R

γ −1; p = (γ −1)ρe; T =

(γ −1)eR

;Assuming Ideal Gas

�

T = −pIWhere we have taken


Incompressible flow


Incompressible flows:

Navier-Stokes equations (conservation of momentum)

�

DρDt

= 0

DρDt

+ ρ∇⋅u = 0

�

∇ ⋅u = 0

∂∂tρu = −∇⋅ ρuu-∇P + ρ fb +∇⋅µ ∇u + ∇Tu( )

�

ρ∂∂tu + ρu ⋅ ∇u = -∇P + ρ fb + µ∇ 2u

For constant viscosity

Computational Fluid DynamicsObjectives:

∂u∂t

+ u∂u∂x

+ v∂u∂y

= −1ρ∂P∂x

+µρ

∂ 2u∂x2

+∂ 2u∂y2

⎛ ⎝ ⎜ ⎞

⎠

∂u∂x

+∂v∂y

= 0

∂v∂t

+ u∂v∂x

+ v∂v∂y

= −1ρ∂P∂y

+µρ

∂ 2v∂x2

+∂ 2v∂y2

⎛ ⎝ ⎜ ⎞

⎠

The 2D Navier-Stokes Equations for incompressible, homogeneous flow:


u ⋅nds = 0S∫

∂∂t

udVV∫ = − uu ⋅n

S∫ dS +ν ∇u ⋅nS∫ dS − 1

ρ pnxS∫ dS

∂∂t

vdVV∫ = − vu ⋅n

S∫ dS +ν ∇v ⋅nS∫ dS − 1

ρ pnyS∫ dS

Incompressibility (conservation of mass)

Navier-Stokes equations (conservation of momentum)


Needed to accelerate/decelerate a fluid particle: Easy to use if viscous forces are small

Needed to prevent accumulation/depletion of fluid particles: Use if there are strong viscous forces

The role of pressure:

Pressure and Advection

Advection:

Newton�s law of motion: In the absence of forces, a fluid particle will move in a straight line


The pressure opposes local accumulation of fluid. For compressible flow, the pressure increases if the density increases. For incompressible flows, the pressure takes on whatever value necessary to prevent local accumulation:

High Pressure Low Pressure

�

∇ ⋅u < 0

�

∇ ⋅u > 0

PressureComputational Fluid Dynamics

Increasing pressure slows the fluid down and decreasing pressure accelerates it

Pressure increases

Pressure decreases

Slowing down Speeding up

Pressure


Diffusion of fluid momentum is the result of friction between fluid particles moving at uneven speed. The velocity of fluid particles initially moving with different velocities will gradually become the same. Due to friction, more and more of the fluid next to a solid wall will move with the wall velocity.

ViscosityComputational Fluid Dynamics

For all but the lowest Reynolds numbers, the fluid flow is unstable and unsteady, forming transient whorls and vortices that greatly increase the transfer of momentum over what viscosity alone can do.

The Kelvin Helmholtz instability of a slip line between flows in different directions is one of the fundamental ways in which steady flow becomes unstable.

Instability and unsteadyness

Time


Nondimensional Numbers—the Reynolds number

Computational Fluid DynamicsNondimensional Numbers

∂u∂t

+ u∂u∂x

+ v∂u∂y

= −1ρ∂P∂x

+µρ

∂ 2u∂x2

+∂ 2u∂y2

⎛ ⎝ ⎜ ⎞

⎠

�

˜ u , ˜ v =u,vU

; ˜ x , ˜ y =x, yL

; ˜ t =UtL

�

U 2

L∂ ˜ u ∂˜ t

+ ˜ u ∂ ˜ u ∂˜ x

+ ˜ v ∂˜ u ∂˜ y

⎛ ⎝ ⎜ ⎞

⎠ = −

1Lρ

∂P∂˜ x

+UL2

µρ

∂ 2 ˜ u ∂˜ x 2

+∂ 2 ˜ u ∂˜ y 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

�

∂˜ u ∂˜ t

+ ˜ u ∂ ˜ u ∂˜ x

+ ˜ v ∂˜ u ∂˜ y

= −L

U 21

Lρ∂P∂˜ x

+L

U 2UL2

µρ

∂ 2 ˜ u ∂˜ x 2

+∂ 2 ˜ u ∂˜ y 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟


�

∂˜ u ∂˜ t

+ ˜ u ∂ ˜ u ∂˜ x

+ ˜ v ∂˜ u ∂˜ y

= −L

U 21

Lρ∂P∂˜ x

+L

U 2UL2

µρ

∂ 2 ˜ u ∂˜ x 2

+∂ 2 ˜ u ∂˜ y 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

�

LU 2

1Lρ

∂P∂˜ x

=1

ρU 2∂P∂˜ x

=∂ ˜ P ∂˜ x

�

LU 2

UL2

µρ

=µ

ULρ=1Re

�

˜ P =PρU 2

�

Re =ρUL

µ

where

where

Nondimensional NumbersComputational Fluid Dynamics

�

∂˜ u ∂˜ t

+ ˜ u ∂ ˜ u ∂˜ x

+ ˜ v ∂˜ u ∂˜ y

= −∂ ˜ P ∂˜ x

+1

Re∂ 2 ˜ u ∂˜ x 2

+∂ 2 ˜ u ∂˜ y 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

�

∂˜ v ∂˜ t

+ ˜ u ∂˜ v ∂˜ x

+ ˜ v ∂ ˜ v ∂˜ y

= −∂ ˜ P ∂˜ y

+1Re

∂ 2 ˜ v ∂˜ x 2

+∂ 2 ˜ v ∂ ˜ y 2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

The momentum equations are:

∂ u∂ x

+∂ v∂ y

= 0

The continuity equation is unchanged

Nondimensional Numbers


Derivation of the equations governing fluid flow in integral form

Conservation of MassConservation of MomentumConservation of Energy

Differential formSummaryIncompressible flowsInviscid compressible flows

OutlineComputational Fluid Dynamics

Integral versus differentialform of the

governing equations


When using FINITE VOLUME approximations, we work directly with the integral form of the conservation principles. The average values of f in a small volume are stored

xj-1/2 xj+1/2

fj+1

f j−1fj

x

f

xj-1 xj xj+1

Finite Volume ApproximationsComputational Fluid Dynamics

∂f∂t+∂F∂x

= 0

∂f∂tdx

Δx∫ +

∂F∂x

Δx∫ dx = 0

ddt

f dxΔx∫ + Fj+1/ 2 − Fj−1/ 2 = 0

To derive an equation that governs the evolution of the average value in each cell, consider:

Integrating the equation over a small control volume of size h= Δx=xj+1-xj

yields: The volume is fixed in space

Finite Volume Approximations


ddt(hf j) = Fj−1/ 2 − Fj+1/ 2

fj =1h

f dxΔx∫

where the average in each cell is defined by:

or:

Fj-1/2 is the flow of f into cell jFj+1/2 is the flow of f out off cell j F is usually called the flux function


Notation:

xj-1/2 xj+1/2

Fj+1/ 2fj

xj

Fj−1/ 2Flow of f into cell j

Flow of f out of cell j

Average value of f in cell j

Center of cell j

Left boundary of cell j Right boundary of cell j



F =Uf

€

F = −D∂f∂x

F =Uf − D∂f∂x

Advection

Diffusion

Advection/Diffusion

The exact form of F depends on the problem:


ddt(hf j) = Fj−1/ 2 − Fj+1/ 2

The statement

Is exactly the original conservation law

Rate of increase in f = inflow of f -outflow of f

The rate of increase of the average value of f is equal to the inflow into the control volume minus the outflow from the control volume



€

∂f∂t

+U∂f∂x

=D∂ 2 f∂x 2

Example: advection/diffusion equation

were

F =Uf − D∂f∂x

∂f∂t+∂F∂x

= 0

Rewrite:

Assume here that U is a constant


xj-1/2 xj+1/2

Fj+1/ 2fj

xj


ddt(hf j) = Fj−1/ 2 − Fj+1/ 2

For each cell

fj =1h

f dxΔx∫

€

Fj+1/ 2 =Uf j+1/ 2 −D∂f∂x j+1/ 2



xj-1/2 xj+1/2

Fj+1/ 2fj

xj


fj+1 / 2 ≈ 12 ( f j+1 + f j)

∂f∂x j+1 / 2

≈fj+1 − fjh

€

Fj+1/ 2 =Uf j+1/ 2 −D∂f∂x j+1/ 2

Approximate:

and

f at a cell boundary gradient of f at a cell boundary

Need to find


€

Fj+1/ 2 =Uf j+1/ 2 −D∂f∂x j+1/ 2

≈12U ( f j+1 + f j) −D

f j+1 − f jh

⎛

⎝ ⎜ ⎞

⎠

Approximate:

€

Fj−1/ 2 =Uf j−1/ 2 −D∂f∂x j−1/ 2

≈12U ( f j + f j−1) −D

f j − f j−1h

⎛

⎝ ⎜ ⎞

⎠

and

ddt

fj ≈1Δt( fj

n+1 − fjn ) time derivative of f



€

hΔt( f jn+1 − f jn ) =

−U (12 ( f j+1 + f j ) − 12 ( f j + f j−1)) +D f j+1 − f j

h−f j − f j−1

h⎛

⎝ ⎜ ⎞

⎠

€

f jn +1 − f j

n

Δt+U2h( f j+1 − f j−1) =

Dh2

f j+1 − 2 f j + f j−1( )

Or, rearranging the terms:

Substituting:

Which is exactly the same as the finite difference equation if we take the average value to be the same as the value in the center of the cell


Example:

U=1; D=0.05; k=1

N=21Δt=0.05

Exact

Numerical



The finite volume point of view has two main advantages over the finite difference point of view

1. Working directly with the conservation principle (no implicit assumption of smoothness and differentiability)

2. Easier visualization of how the solution is updated (the fluxes have a physical meaning)

Analysis of accuracy and stability is, however, usually easier using the finite difference point of view


Most physical laws are based on CONSERVATION principles: In the absence of explicit sources or sinks, f is neither created nor destroyed.

For a control volume fixed in space, we can state the conservation of f as:


Amount of f in a control volume after a given time interval Δt

Amount of f that flows into the control volume during Δt

Amount of f that flows out of the control volume during Δt

Amount of f in the control volume at the beginning of the time interval Δt

= + -

Computational Fluid DynamicsFinite Volume Approximations

Rate of increase of f in a control volume during a given time interval Δt

Net rate of flow of f into the control volume during the time interval Δt

=

This can be restated in rate form as:

The mathematical equivalent of this statement is:

ddt

f dv = − fu ⋅nCS!∫CV∫ ds

While we can derive a partial differential expression from this equation, often it is better to work directly with the conservation principle in integral form

conservation mass - university of notre damegtryggva/cfd-course2017/lecture-4...the conservation law...

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