Governing Equations

Fluid Mechanics Foundations (1)

Outline• Introduction to Governing Equations• Derivation of Governing Equations

– The Continuity Equation– Conservation of Momentum– The Energy Equation

• Boundary Conditions• The Gas Dynamics Eq. and the Full Potential Eq.• Special Cases• Which governing equation should be used ?• Mathematical Classification• Requirements for a Complete Problem Formulation

Introduction to Governing Equations

• The fluid is defined by :– an equation of state

• p = ρ R T

– the thermodynamic and transport properties• the ratio of specific heats, γ• viscosity, μ,• the coefficient of heat conduction, k.

Introduction to Governing Equations

• The motion of the fluid is controlled by– governing equations

– boundary conditions

• The governing equations are given by conservation laws:– mass continuity– momentum Newton’s 2nd Law, F=ma– energy 1st Law of Thermodynamics

Introduction to Governing Equations

• Coordinate systems– Cartesian coordinates are normally used to

describe vehicle geometry.– General nonorthogonal curvilinear coordinates

Introduction to Governing Equations

• In general Cartesian coordinates, the independent variables are x, y, z, and t.

• We want to know the velocities, u, v, w, and the fluid properties; p, ρ, T.

• These six unknowns require six equations.– continuity 1 equation– momentum 3 equations– energy 1 equation– equation of state 1 equation

Derivation of Governing Equations

• Motivations– We want to find the flowfield velocity, pressure

and temperature distributions.

– We need to develop a mathematical model of the fluid motion suitable for use in numerical calculations.

– The mathematical model is based on the conservation laws and the fluid properties.

Derivation of Governing Equations

• Approaches– Lagrangian

• Each fluid particle is traced as it moves around the body. • This method corresponds to the conventional concept of Newton’s

Second Law.

– Eulerian• We look at the entire space around the body as a field, and

determine flow properties at various points in the field while the fluid particles stream past.

• We consider the distribution of velocity and pressure throughoutthe field, and ignore the motion of individual fluid particles.

Derivation of Governing Equations

• The form of Equations– The differential form

• Most frequently used in fluid mechanics analysis

– The integral form• Numerically, integral more accurately computed

than derivatives

The Continuity Equation• The statement of conservation of mass is in words simply:

net outflow of mass decrease of massthrough the surface = within thesurrounding the volume volume

Control volume for conservation of mass

YXt

decreasemassofchangeinYoutYinXoutX

ΔΔ∂∂

=

=−−−+−−−

ρ)(

][][][][

The Continuity Equation• The differential form

• The integral form

2-D

3-D

Vector form

0=∂∂

+∂∂

+∂∂

+∂∂

zw

yv

xu

tρρρρ

0=∂∂

+∂∂

+∂∂

yv

xu

tρρρ

0)( =⋅∇+∂∂ Vρρ

t

∫∫∫∫∫ ∂∂

−=⋅∧

V

dVt

dSn ρρV

The Continuity Equation

• Comments– The equation is for all flows, viscous or

inviscid, compressible or incompressible.

– Using the Gauss Divergence Theorem, two equations are related:

0)( =⎥⎦⎤

⎢⎣⎡

∂∂

+⋅∇∫∫∫ VV dtV

ρρ

Conservation of Momentum

• Newton’s 2nd Law:

The time rate of change of momentum of a body equals the net force exerted on it.

• For a fixed mass this is the famous equation

DtDmm VaF ==

Conservation of Momentum• Substantial Derivative

– We need to apply Newton’s Law to a moving fluid element from our fixed coordinate system.

– Consider any fluid property, Q(r,t).

Moving particle viewed from a fixed coordinate system

The change in position of the particle between r at t, and r+Δr at t+Δt is

),(),(),(),(tQtttQ

tQttsQQrVr

rr−∇+Δ+=

−∇+Δ+=Δ

The rate of change of Q

ttQtttQ

tQ

DtDQ

tt Δ−Δ+Δ+

=ΔΔ

=→Δ→Δ

),(),(limlim00

rVr

Conservation of Momentum

• The rate of change is in two parts, one for a change in time, and one for a change in space.

• We write the change of Q as a function of both time and space using the Taylor series expansion as:

ttQtttQ

tQ

DtDQ

tt Δ−Δ+Δ+

=ΔΔ

=→Δ→Δ

),(),(limlim00

rVr

, ,

( , ) ( , )t t

Q QQ t t t Q t t V tt s

∂ ∂+ Δ + Δ = + Δ + + Δ +

∂ ∂r r

r V r

(1)

(2)

Substitute (1) into (2), and take the limit, we obtain:

Conservation of Momentum

– The second term has the unknown velocity V multiplying a term containing the unknown Q. This is important !

– The convective derivative introduces a fundamental nonlinearity into the system.

VsQ

tQ

tQ

t ∂∂

+∂∂

=ΔΔ

→Δ 0lim

Local time derivative, local derivative

Variation with change of position, convective derivative

Substantial derivative

Conservation of Momentum

– In coordinates, V = {u, v, w}, and the substantial derivative becomes:

tww

twv

twu

tw

DtDw

tvw

tvv

tvu

tv

DtDv

tuw

tuv

tuu

tu

DtDu

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

Conservation of Momentum• Forces

– body forces– pressure forces– shear forces

Details of forces acting on a two-dimensional control volume

Conservation of Momentum

– The net force in the x-direction is found to be:

– Using the substantial derivative and the definition of the mass, m = ρ Dx Dy Dz, and considering the x component, max = Fx

xyx

yxx

yfx yxxxx ΔΔ∂∂

+ΔΔ∂∂

+ΔΔ⋅ )()( ττρ

zxyz

zxyy

zyxx

zyxDtDuzyx

zx

yxxx

ΔΔΔ∂∂

+

ΔΔΔ∂∂

+ΔΔΔ∂∂

+ΔΔΔ⋅=ΔΔΔ⋅

)(

)()(

τ

ττρρ

Conservation of Momentum

• General conservation of momentum relations

• Comments• They are valid for anything !

zyxf

DtDw

zyxf

DtDv

zyxf

DtDu

zzyzxzz

zyyyxyy

yxyxxxx

∂∂

+∂

∂+

∂∂

+=

∂

∂+

∂

∂+

∂

∂+=

∂

∂+

∂

∂+

∂∂

+=

τττρρ

τττρρ

τττρρ

Conservation of Momentum• Newtonian fluid

– Stress is a linear function of the rate of strain

• The set of assumptions1. The stress-rate-of-strain relations must be independent of

coordinate system.

2. When the fluid is at rest and the velocity gradients are zero (the strain rates are zero), the stress reduces to the hydrostatic pressure.

3. The assumption between viscosity coefficients (Stoke’sHypothesis).

yu∂∂

= μτ

Conservation of Momentum

• Relations between stress and μ based on the assumptions.

and

xwp

yvp

xup

zz

yy

xx

∂∂

+⋅∇−−=

∂∂

+⋅∇−−=

∂∂

+⋅∇−−=

μμτ

μμτ

μμτ

232

232

232

V

V

V

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

==

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

==

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

==

yw

yv

xw

zu

xv

yu

zyyz

zxxz

yxxy

μττ

μττ

μττ

Conservation of Momentum• The classic Navier-Stokes Equations

– written in the standard aerodynamics form neglecting the body force.

– Non-linear ( recall that superposition of solutions is not allowed)– Highly coupled– Long !

⎟⎠⎞

⎜⎝⎛ ⋅∇−

∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

∂∂

+∂∂

−=

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛⋅∇−

∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+∂∂

−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛ ⋅∇−

∂∂

∂∂

+∂∂

−=

V

V

V

μμμμρ

μμμμρ

μμμμρ

322

322

322

zw

zyw

zv

yzu

xw

xwp

DtDw

zv

yw

zyv

yxv

yu

xyp

DtDv

zu

xw

zxv

yu

yxu

xxp

DtDu

Conservation of Momentum

• Euler Equations– When the viscous terms are small, and thus ignored, the

flow is termed inviscid.

– The resulting equations are known as the Euler Equations.

0

0

0

=∂∂

+∂∂

+∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

+∂∂

zp

zww

ywv

xwu

tw

yp

zvw

yvv

xvu

tv

xp

zuw

yuv

xuu

tu

ρ

ρ

ρ

Conservation of Momentum

• The integral formulations of the equations– The momentum change, ρV, is proportional to the

force.

– This statement can also be converted to the differential form using the Gauss Divergence Theorem.

surfacevolumeV

dvt

dsn FFFVVV +==∂∂

+⋅ ∫∫∫∫∫∧

ρρ )(

Conservation of Momentum

• Comments on N-S equations:– The derivation of the Navier-Stokes Equations is for general

unsteady fluid motion.

– Because of limitations in our computational capability, these equations are for laminar flow.

– When the flow is turbulent, the usual approach is to Reynolds-averaged the equations, with the result that additional Reynolds stresses appear in the equations.

• The addition of new unknowns requires additional equations.

• This problem is treated through turbulence modeling and is discussed in turbulence models.

The Energy Equation• 1st Law of Thermodynamics

– The sum of the work and heat added to a system will equal the increase of energy.

– For the fixed control volume coordinate system, the rate of change is:

where

e is the internal energy per unit mass

The last term is the potential energy, i.e. the body force.

(1)

WQdEt δδ +=Change of total energy of the system

Change of heat added Change of work done on the system

••

+= WQDt

DEt ⎟⎠⎞

⎜⎝⎛ ⋅−+= rg2

21 VeEt ρ

The Energy Equation– In aerodynamics the potential energy is neglected. – Et can also be written in terms of specific energy as:

221

00 VeewhereeEt +== ρ

• The derivation of the energy equation– To obtain the energy equation we need to write the Q and W in

terms of flow properties.

– Using the concept of control volume, we can obtain:

)(.

TkqQ ∇⋅+∇=⋅−∇=

(2)

The rate of change of Q

Work done on a control volume

(4))()( yyyxxyxx vuy

vux

divW ττττ +∂∂

++∂∂

=−=•

w

– Substituting Eqs.(2) ,(3) and (4) into (1) , we obtain:

wdivTkDt

VeD−∇⋅∇=

+)(

)( 221ρ

– Substituting in the relations for theτ’s in terms of μand the velocity gradients

– Introducing the definition of enthalpy, h = e + p/ρ,

– We obtain a the classical energy equation

where

Φ+∇⋅∇=− )( TkDtDp

DtDhρ

Heat conduction Viscous dissipation (always positive)

Review on Governing Equations

• The Continuity Equation • The N-S equation (Conservation of Momentum )

0=∂∂

+∂∂

+∂∂

+∂∂

zw

yv

xu

tρρρρ

⎟⎠⎞

⎜⎝⎛ ⋅∇−

∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

∂∂

+∂∂

−=

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛⋅∇−

∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+∂∂

−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛ ⋅∇−

∂∂

∂∂

+∂∂

−=

V

V

V

μμμμρ

μμμμρ

μμμμρ

322

322

322

zw

zyw

zv

yzu

xw

xwp

DtDw

zv

yw

zyv

yxv

yu

xyp

DtDv

zu

xw

zxv

yu

yxu

xxp

DtDu

• The Energy Equation

Φ+∇⋅∇=− )( TkDtDp

DtDhρ

• The equation of statep = ρ R T

Boundary Conditions

• Introduction

– Boundary conditions are the means through which the solution of the governing equations produce differing results for different situations.

– In computational aerodynamics the specification of boundary conditions constitutes the major part of any effort.

– The application of the software usually requires the user to specify the boundary conditions.

Boundary Conditions• The condition on the surface

– For an inviscid steady flow over solid surface

V • n = 0 non-penetration condition

– If the flow is viscous the statement becomes even simpler:

V = 0 no-slip condition

– If the surface is porous, and there is mass flow, the values of the surface velocity must be specified

– Numerical solutions of the Euler and N-S solutions require that other boundary conditions be specified.

• conditions on pressure and temperature are required

Boundary Conditions

• Conditions away from the body– Commonly this means that at large distances from

the body the flowfield must approach the freestream conditions.

– Questions • How far away is infinity ?

• Exactly how should you specify the farfield boundary condition numerically?

– How to best handle these issues is the basis for many papers currently appearing in the literature.

Boundary Conditions

• Another use of the boundary conditions

– a means of modeling physics

• Kutta Condition: (for example)– The viscous effects at the trailing edge can be

accounted for in an inviscid calculation without treating the trailing edge problem explicitly.