review of the navier-stokes equations

8/22/2019 Review of the Navier-Stokes Equations

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AML 811

Lecture 11

Review of the Navier-Stokes Equations


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Recap : Modified Differential Equation

with FTBS for wave equation The MDE for the FTBS for wave equation can be written as

DissipationDispersion

Both the dissipation and dispersion terms are zero for the CFL

number, c = 1 This happens for the higher order terms also. Hence, FTBS is exact

for a CFL for 1 with the wave equation

Note that as c decreases, the MDE has a higher coefficient of

dissipation. This is what was reflected in our simulations

See Hoffmann and Chiang, Vol 1, Appendix C for the algebra in all its glory


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Recap: FTBS for wave equation with CFL

= 1

Gives the exact solution because

FTBS has the same form as the exact solution for CFL = 1

All the truncation error terms in the MDE cancel for CFL = 1


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= 0.5

Solution diffusive because of the dissipative term in the MDE is notzero. Note that the MDE is still consistent as truncation error goesto zero as grid size decreases


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= 0.9

Solution diffusive because of the dissipative term in the MDE is notzero.

Since 1-c is smaller than that for CFL = 0.5, solution is lessdissipative for this CFL.


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Recap: Anti-diffusive behavior: CFL =

1.1Artificial Dissipation coefficient

Solution anti-diffusive because of the coefficient of artificialdissipation is negative. (Never, never confuse this for the wholedissipation term itself)

This gives additional significance to why the CFL 1 for stability


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Recap: Upwind methods

FTBS works while FTFS does not because FTBS

takes numerical information consistent where the

direction of the characteristics

Such schemes are known as upwind schemes as

they take information only in a direction upwind of

the wave propagation. For e.g. Higher order upwind method for

x

ua

t

u

=

+=

+

x

uuua

t

uun

i

n

i

n

i

n

i

n

i

2

4321

1

What would be first order upwind scheme for

0=

+

x

ua

t

u

01

1

=

+

++

x

uua

t

uun

i

n

i

n

i

n

i

0


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Recap: Explicit Method: Lax-Wendroff

The original equation is used in the Taylor

series expansion to obtain the finite

difference equation. Hence, spatial andtemporal derivatives are connected

Second order accurate in space and time Stability for c 1


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Recap: Lax-Wendroff CFL = 1

Blue lines : Exact

Red lines : Numerical

Exact solution for CFL = 1 for the same

reasons as FTBS


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Recap: Lax-Wendroff CFL = 0.9

Blue lines : Exact


Solution is oscillatory, unlike FTBS

Solution shows both dissipation and dispersion


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Blue lines : Exact



Solution shows both dissipation and dispersion.

More of all non-idealities than CFL = 0.9


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Blue lines : Exact



Solution shows both dissipation and dispersion.

More of all non-idealities than CFL = 0.9


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Recap: Explicit Method: Lax-Wendroff

Can be viewed as addition of artificial dissipation toFTCS.

For CFL < 1, the dissipation is not enough to kill allthe oscillations caused by the central scheme.(When would the scheme be the same as FTBS?)

Since the scheme is second order accurate, thetruncation error term starts with a third orderderivative which explains the dispersion


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Recap: Some Implicit Methods for the wave

equation 02

1

1

1

1

1

=

+

++

++

x

uua

t

uun

i

n

i

n

i

n

i

BTCS

0

11

1

1

=

+

++++

x

uua

t

uun

i

n

i

n

i

n

i

BTBS

0222

11

1

1

1

1

1

=

+

+

++

+

++

x

uu

x

uua

t

uun

i

n

i

n

i

n

i

n

i

n

i

Crank Nicholson

The stability restriction on the time step explicit schemes for the wave equationis proportional to the grid size unlike the parabolic equation where is it isproportional to x2

Since we require accurate solutions in time, the time step is already restrictedeven for implicit schemes by the accuracy constraint

Often, we dont gain anything by using implicit schemes for the wave equation. Explicit schemes are hence far more used for hyperbolic problems and for the

Navier-Stokes, the convective terms are often computed explicitly.


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Summary of Lecture 10

Modified Differential Equation (MDE) for

FTBS shows why it is increasingly dissipativeas CFL decreases

FTBS is a first-orderupwind method. Higher

order upwind methods can also be derived

Lax-Wendroff : Explicit scheme which is

second accurate in time and space. However,it adds non-physical oscillations and is also

dispersive


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Main ideas and terms in the course till now

Basic properties of PDEs Hyperbolic, parabolic and elliptic PDEs

Characteristics

Deriving finite difference formulas of any order for auniform and non-uniform stencil

Taylor series table, polynomial fit

Convergence = Consistency + Stability

Modified Differential Equation (MDE)

Von Neumann Stability criterion

Explicit vs Implicit Methods


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Main ideas and terms in the course till now

Methods for Parabolic Equations FTCS (explicit), Crank Nicholson (Implicit)

Diffusion number

Approximate Factorization Methods :ADI, Fractional-Step

Methods for Elliptic Equations 5-point stencil and 9-point stencil

Solution of linear system of equations Direct vs iterative methods

Jacobi, Gauss-Seidel, SOR Point and line methods : Line Gauss-Seidel, LSOR

Methods for Hyperbolic equations Upwind schemes, numerical dissipation

Lax-Wendroff Dispersion and dissipation due to numerical method


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Assignments,Minors,Projects, etc

Assignments Next assignment will be given tomorrow. Please submit the

previous assignment by then

No late assignments from now on

Minors Minor 2 will be a take home exam i.e. effectively a (longish!)

homework except that no discussions would be allowed.

Final Project Will not be a long coding exercise but analysis of a subject and a

few papers. Please give me a subject of your interest and I canfind something of interest to you (by this Friday). If you do nothave anything particular you are interested in, I can assign you

something by myself.Etc

Please fill in the mid-term evaluations and return them to me


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Assignments,Minors,Projects, etc

Assignments Next assignment will be given tomorrow. Please submit the

previous assignment by then

No late assignments from now on

Minors Minor 2 will be a take home exam i.e. effectively a (longish!)

homework except that no discussions would be allowed.

Final Project Will not be a long coding exercise but analysis of a subject and a

few papers. Please give me a subject of your interest and I canfind something of interest to you (by this Friday). If you do nothave anything particular you are interested in, I can assign you

something by myself. Etc

Please fill in the mid-term evaluations and return them to me


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The governing equations of fluid flow

Governing equations can be written in many forms Differential, integral, conservative, non-conservative

All of them express the same physical law but oneform or the other can be more correct for numericalapproximation.

The laws of physics a fluid obeys at every point Conservation of mass.

Newtons second Law. F = ma

Conservation of energy

A solid obeys these too. So what distinguishes afluid from a solid?


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Concept of a fluid What distinguishes a fluid from a solid?

A fluid deforms continuously under anapplied shear force (i.e. a forcetangential to its surface)

Hence, on a static free surface willalways be perpendicular to the netforce.

Note: A fluids resistance depends on

rate of strain (i.e. rate of deformation)instead of strain.

Linear elastic solid => Stress isproportional to strain

Newtonian fluid => Stress isproportional to rate-of-strain

Net force

Net force is normal to the

static free surface


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The continuum hypothesis

A fluid is made of atoms and molecules so whatdo density, pressure, velocity, etc at a pointmean?

In theory: We assume that the fluid is actually acontinuum i.e. that it is continuous medium andevery point in space has a fluid particle in it.

In practice,

Density and other quantities are molecularaverages over a volume which is big enough tocontain sufficient molecules but small enough notto include macroscopic variations


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Different ways of deriving the governing

equations Control Volume analysis

Concentrate on a finite volume of fluid and carefully apply conservationprinciples

Basic idea : Change of quantity inside volume = Flux of quantity in Flux ofquantity out

Results in integral form of equations

Can derive finite volume methods using this idea

This form holds true regardless of discontinuities such as shocks, etc

Differential AnalysisConcentrate on the dynamics of an infinitesimal fluid element and apply the

physical conservation laws

Results in differential form of equations

Can derive finite difference methods from this form of equations

The equation does not hold through discontinuities as derivatives are notdefined


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Different ways of deriving the governing

equations Control Volume analysis

Concentrate on a finite volume of fluid and carefully apply conservationprinciples

Basic idea : Change of quantity inside volume = Flux of quantity in Flux ofquantity out

Results in integral form of equations

Can derive finite volume methods using this idea

This form holds true regardless of discontinuities such as shocks, etc

Differential Analysis Concentrate on the dynamics of an infinitesimal fluid element and apply the

physical conservation laws

Results in differential form of equations

Can derive finite difference methods from this form of equations

The equation does not hold through discontinuities as derivatives are notdefined


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Differential Analysis : The substantial,

total or material derivative

Two perspectives of looking at flow Eulerian : Looking at a fixed point in space

Lagrangian : Following an individual material particle.

Physical laws are known for individual particles but we areinterested at a fixed point in space.

The connection between the two ideas: The substantialderivative


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Taylor series


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Dt

D

dt

d

Chain rule and physical

arguments give the same result


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Conservation of mass

Mass is conserved =>

Rate of change of mass in volume = Inlet mass flux Outgoing mass flux

Coordinate independentform

M i


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Momentum equation

To give an idea of an alternate approach (the previous approach wouldgive the same result), consider a moving infinitesimal fluid element

Types of external forces acting on the fluid element Body forces : Act directly on the volume of fluid element (i.e. at each point

inside a volumetric element). These are forces that act at a distance e.g :Gravitational, Electromagnetic etc.

Surface forces : These forces act only on the exposed surfaces of anelement. These act locally only. Physically, they are due to two sources Pressure on the surface imposed due to the external fluid surrounding the element

Viscous forces caused by the external fluid dragging, etc on the fluid

Dt

vDa

r

r

=Use

S f f h fl id l


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Surface forces on the fluid element

Only forces in the x-direction shown

Note the two types of surface forces here Pressure on the surface imposed due to the external fluid surrounding the

element. Pressure is purely a normal force

Viscous forces caused by the external fluid dragging, etc on the fluid.Viscous forces can be both normal and shear

S f f h fl id l


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Surface forces on the fluid element

Note the two types of surface forces here Pressure on the surface imposed due to the external fluid surrounding the

element. Pressure is purely a normal force

Viscous forces caused by the external fluid dragging, etc on the fluid.Viscous forces can be both normal and shear

Viscous shear force

Viscous normal force

F b l h fl id l


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Force balance on the fluid element

Body force per

unit mass

External force = Surface + Body

F h fl id l


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F = ma on the fluid elementExternal force = Surface + Body

Momentum equation in x-direction

F th fl id l t


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F = ma on the fluid element

Momentum equation in the y-direction

Momentum equation in the z-direction

Newtonian fl id and the Stokes hypothesis


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Newtonian fluid and the Stokes hypothesis

The momentum equations have viscous stresses asunknowns. To close the system of equations we

need some relationship between the stress and thevelocity derivatives

The assumption of a Newtonian fluid (Stress is

proportional to rate-of strain) provides closure. Thisassumption is not valid for non-Newtonian fluidssuch as blood etc.

The previous assumption still leaves the bulkviscosity coefficient unknown. Stokes hypothesisrelates the molecular viscosity to the bulk viscosity.



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Newtonian fluid

Stokes hypothesis

The Navier Stokes momentum equations


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The Navier-Stokes momentum equations

Coupled system of nonlinear

PDEs

Energy equation


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Energy equation

Uses first law of thermodynamics

For closure, assume

Fouriers law of heat conduction (to relate heat

flux to temperature)

Ideal gas law : To relate temperature, pressure

and density

Energy balance on an element


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Energy balance on an element

The energy equation in conservation form


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The energy equation in conservation form

Equations of motion


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Equations of motion

System of coupled, non-linearPDEsContinuity (mass)

Momentum

Energy

Summary


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Summary

Concept of a fluid: Deforms continuously under shear

Continuum hypothesis

Eulerian and Lagrangian viewpoints

Conservation equations Continuity: Mass is conserved

Momentum : F = ma

Energy. First law of thermodynamics

These laws + some constitutive laws give the full equations forflow

The full equations can be simplified for various cases : inviscid,incompressible, boundary layers, low Reynolds number, etc. The

simplified equations often also exhibit purely hyperbolic, elliptic,parabolic etc.

review of the navier-stokes equations

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