Governing Equations

Fluid Mechanics Foundations (2)

Outline• Introduction to Governing Equations• Derivation of Governing Equations

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

• Boundary Conditions• A Review of the Governing Equations• The Gas Dynamics Eq. and the Full Potential Eq.• Special Cases• Which governing equation should be used ?• Requirements for a Complete Problem Formulation

A Review of the Governing Equations

• The Continuity Equation • The N-S equations (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

A Review of the Governing Equations

• Euler Equations– When the flow is termed inviscid

0

0

0

=∂

∂+

∂∂

+∂∂

+∂∂

+∂∂

=∂

∂+

∂∂

+∂∂

+∂∂

+∂∂

=∂

∂+

∂∂

+∂∂

+∂∂

+∂∂

zp

zww

ywv

xwu

tw

yp

zvw

yvv

xvu

tv

xp

zuw

yuv

xuu

tu

ρ

ρ

ρ

The Gas Dynamics Equation

• Introduction– For inviscid flow it is useful to combine the equations in

a special form known as the gas dynamics equation.

– This equation is used to obtain the “full” nonlinear potential flow equation. Many valuable results can be obtained using the potential flow approximation.

– The equation is valid for any flow assumed to be inviscid.

– The starting point for the derivation is the Euler equations, the continuity equation and the equation of state.

The Gas Dynamics Equation

• Derivation for 2-D steady flowUse a thermodynamic definition to rewrite the pressure term in the momentum equation

yp

yp

xp

xp

∂∂

∂∂

=∂∂

∂∂

∂∂

=∂∂ ρ

ρρ

ρ

Use the definition of the speed of soundρ∂

∂=

pa2

∂p/ ∂x, ∂p/ ∂y can be written as

ya

yp

xa

xp

∂∂

=∂∂

∂∂

=∂∂ ρρ 22

• Derivation for 2-D steady flow

Recall Euler equations for 2-D steady flow

u times x momentum

v times y momentumy

pyvv

xvu

xp

yuv

xuu

∂∂

−=∂∂

+∂∂

∂∂

−=∂∂

+∂∂

ρ

ρ1

1

Use modified ∂p/ ∂x, ∂p/ ∂y expressions in above equations

ypv

yvv

xvvu

xpu

yuuv

xuu

∂∂

−=∂∂

+∂∂

∂∂

−=∂∂

+∂∂

ρ

ρ

2

2

yau

ypv

yvv

xvvu

xau

xpu

yuuv

xuu

∂∂

−=∂∂

−=∂∂

+∂∂

∂∂

−=∂∂

−=∂∂

+∂∂

ρρρ

ρρρ2

2

22

• Derivation for 2-D steady flow

Add those two equations above

yav

ypv

yvv

xvvu

xau

xpu

yuuv

xuu

∂∂

−=∂∂

−=∂∂

+∂∂

+∂∂

−=∂∂

−=∂∂

+∂∂

ρρρ

ρρρ

22

22

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

yv

xua

yvv

xvvu

yuuv

xuu ρρ

ρ

222

• Derivation for 2-D steady flow

From the continuity equation for 2-D steady flow

0=∂

∂+

∂∂

yv

xu ρρ

Expand it

0=∂∂

+∂∂

+∂∂

+∂∂

yv

yv

xu

xu ρρρρ

yv

xu

yv

xu

∂∂

−∂∂

−=∂∂

+∂∂ ρρρρ

• Derivation for 2-D steady flowCombine the modified momentum equation based on Euler equations with the modified continuity equation

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

+∂∂

yv

xua

yvv

xvvu

yuuv

xuu ρρ

ρ

222

Finally, we obtain the gas dynamics equation

0)()( 2222 =∂∂

−+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−yvav

xv

yuuv

xuau

The modified momentum equation

The modified continuity equation

yv

xu

yv

xu

∂∂

−∂∂

−=∂∂

+∂∂ ρρρρ

The Gas Dynamics Equation

• The Gas Dynamics Equation in three dimensions:

• Comments The gas dynamics equation is derived from Euler equations, the continuity equation and the equation of state.

Only one equation.

The equation contains only u,v,w and a.

0)()()( 222222 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−+∂∂

−+∂∂

−yw

zvuv

zu

xwuw

xv

yuuv

zwaw

yvav

xuau

The Gas Dynamics Equation

• The Gas Dynamics-Related Energy Equation – The special form of the energy equation in two dimensions:

))(2

1( 2220

2 vuaa +−

−=γ

– in three dimensions:

))(2

1( 22220

2 wvuaa ++−

−=γ

– Derivation can be found from the text

2 2 20

12

a const a Uγ∞ ∞

−= = +where

Full Potential Equation

• Assume that the flow be irrotational.

• This is valid for inviscid flow when the onset flow is uniform and there are no shock waves.

• The irrotational flow assumption is stated mathematically as :

curl V = 0

• V can be defined as the gradient of a scalar quantity

Φ∇=V

Full Potential Equation

– The velocity components are u = Φx , v = Φy and w = Φz

– Using the gas dynamics equation, the non-linear or “full” potential equation is then:

– Comments– The classic form– A single partial differential equation– nonlinear

0222)()()( 222222 =ΦΦΦ+ΦΦΦ+ΦΦΦ+Φ−Φ+Φ−Φ+Φ−Φ zxxzyzzyxyyxzzzyyyxxx aaa

Full Potential Equation

• Equivalent Divergence Form and Energy Equation– Divergence form for continuity equation in two dimensions

))(2

1( 2220

2 vuaa +−

−=γ

– Energy equation

0)()( =Φ∂∂

+Φ∂∂

yx yxρρ

• Comments• This form is used in most computational fluid dynamics codes.• The full potential equation is still nonlinear

11

)])(11(1[ 22 −Φ+Φ

+−

−= γ

γγρ yx

0=∂

∂+

∂∂

yv

xu ρρ

Special Cases• Motivations

– The simplified forms of the equations is able to provide explicit physical insight into the flowfield process, and has played an important role in the development of aerodynamic concepts.

– The simplified equations are easy to be solved.

• Assumption: small disturbance– We expect this assumption to be valid for inviscid flows over

streamlined shapes.

– These ideas are expressed mathematically by small perturbation or asymptotic expansion methods.

Special Cases:Small Disturbance Form of the Energy Equation

• The expansion of the simple algebraic statement of the energy equation provides an example of a small disturbance analysis

))(2

1( 2220

2 vuaa +−

−=γ

'22 2)2

1( uUaa ∞∞ ⋅−

−=γ

Letting u = U∞+ u', v = v’

u ’ << U ∞ , v’ < < U∞

This is a linear relation between the disturbance velocity and the speed of sound

012''

≈⎟⎟⎠

⎞⎜⎜⎝

⎛⇒<<

∞∞ Uu

Uu

2'2''22 22

1 vuuUaa ++⎟⎠⎞

⎜⎝⎛ −

−= ∞∞γ

Neglect as small

Special Cases:Small Disturbance Expansion of the Full Potential Equation

• The full potential equation given above (in 2D for simplicity):

0)(2)( 2222 =Φ−Φ+ΦΦΦ+Φ−Φ yyyxyyxxxx aa

• The velocity as a difference from the freestream velocity. Introduce a disturbance potential φ, defined by:

yy

xx

vUu

yxxU

φφ

φ

==Φ+==Φ

+=Φ

∞

∞ ),( where we have introduced a directional bias. The x- direction is the direction of the freestream velocity.

We will assume that φx and φ y are small compared to U∞.

Special Cases:Small Disturbance Expansion of the Full Potential Equation

• The full potential equation became

• Comments• where the φx

2, φ y2 terms are neglected in the coefficients.

• This equation is still nonlinear, but is in a form ready for thefurther simplifications described below.

0)1(112)1(1 2222 =⎥⎦

⎤⎢⎣

⎡−+−+⎟⎟

⎠

⎞⎜⎜⎝

⎛++⎥

⎦

⎤⎢⎣

⎡++−

∞∞

∞∞∞

∞∞∞ yy

yxy

yxxx

x

UM

UUM

UMM φ

φγφ

φφφφγ

Special Cases:Transonic Small Disturbance Equation

• Introduction– Transonic flows contain regions with both subsonic and

supersonic velocities.

– Any equation describing this flow must simulate the correct physics in the two different flow regimes.

• The essential nonlinearity of transonic flow– The rapid streamwise variation of flow disturbances in the x-

direction, including normal shock waves.

yx ∂∂

>∂∂

Special Cases:Transonic Small Disturbance Equation

• Equation– The transonic small disturbance equation retains the key term in the

convective derivative u(∂u/ ∂ x), which allows the shock to occur in the solution.

0])1()1[( 22 =++−−∞

∞∞ yyxxx

UMM φφ

φγ

• Comments– It is still nonlinear, and can change mathematical type– It is valid for transonic flow– Can be solved on your personal computer

0)1(112)1(1 2222 =⎥⎦

⎤⎢⎣

⎡−+−+⎟⎟

⎠

⎞⎜⎜⎝

⎛++⎥

⎦

⎤⎢⎣

⎡++−

∞∞

∞∞∞

∞∞∞ yy

yxy

yxxx

x

UM

UUM

UMM φ

φγφ

φφφφγ

Special Cases:Prandtl-Glauert Equation

• When the flowfield is entirely subsonic or supersonic, all terms involving products of small quantities can be neglected in the small disturbance equation.

0)1( 2 =+− ∞ yyxxM φφ

• Comments– This is a linear equation

– Valid for small disturbance flows that are either entirely supersonic or subsonic.

– For subsonic flows this equation can be transformed to Laplace’s Equation, while at supersonic speeds this equation takes the form of a wave equation.

0])1()1[( 22 =++−−∞

∞∞ yyxxx

UMM φφ

φγ

Special Cases:Laplace’s Equation

• Assumption– Assuming that the flow is incompressible, ρ is a constant and can

be removed from the modified continuity equation

0=+ yyxx φφ0)()( =Φ∂∂

+Φ∂∂

yx yxρρ

• Comments– When the flow is incompressible, this equation is exact when

using the inviscid irrotational flow model.

– Does not require the assumption of small disturbances

Continuity Eq.

Summary on Governing EquationsThe connection between various flowfield models

Which governing equation should be used ?

• For high Reynolds number attached flow, the pressure can be obtained very accurately without considering viscosity.

• If the onset flow is uniform, and any shocks are weak, Mn < 1.25 or 1.3, then the potential flow approximation is valid.

• When shocks begin to get strong and are curved, the solution of the complete Euler equations is required.

• If a slight flow separation exists, a special approach using theboundary layer equations can be used interactively with the inviscidsolution to obtain a solution.

Which governing equation should be used ?

• If flow speed is low, the flow can be considered as incompressible, and Laplace’s Equation is valid.

• If flow is subsonic or supersonic with small disturbance, Prandtl-Glauert Equation is valid.

• If flow is transonic with small disturbance, TSDE is valid.

• When significant separation occurs, or you cannot figure out thepreferred direction to apply a boundary layer approach, the Navier-Stokes equations are used.

Requirements for a Complete Problem Formulation

• Governing Equations

• Boundary Conditions

• Coordinate System Specification

If this is done, then the mathematical problem being solved is considered to be well posed.

Homework 2

• Derive the N-S equations from the conservation law of momentum.

• Describe the connections among the various governing equations, including – N-S equations

– Euler equations

– Potential or Full Potential equation

– Transonic Small Disturbance equation

– Prandtl-Glauert equation

– Laplace's equations