Applied Gas Dynamics Compressible Flow Equations - … · Applied Gas Dynamics Compressible Flow Equations Ethirajan Rathakrishnan Applied Gas Dynamics, John Wiley & Sons (Asia) Pte

Applied Gas Dynamics

Compressible Flow Equations

Ethirajan Rathakrishnan

Applied Gas Dynamics, John Wiley & Sons (Asia) Pte Ltd c© 2010 Ethirajan Rathakrishnan 1 / 90

Introduction

The one-dimensional analysis given in the earlier chapters are validonly for flow through infinitesimal streamtubes. In many real flow situa-tions, the assumption of one-dimensionality for the entire flow is at bestan approximation. In problems like flow in ducts, the one-dimensionaltreatment is adequate. However, in many other practical cases, theone-dimensional methods are neither adequate nor do they provide in-formation about the important aspects of the flow.


For example, in the case of flow past the wings of an aircraft, flowthrough the blade passages of turbine and compressors, and flow throughducts of rapidly varying cross-sectional area, the flow field must bethought of as two-dimensional or three-dimensional in order to obtainresults of practical interest.


Because of the mathematical complexities associated with the treat-ment of the most general case of three-dimensional motion – includingshocks, friction and heat transfer, it becomes necessary to conceivesimple models of flow, which lend themselves to analytical treatmentbut at the same time furnish valuable information concerning the realand difficult flow patterns. We know that by using Prandtl’s boundarylayer concept, it is possible to neglect friction and heat transfer for theregion of potential flow outside the boundary layer (see Section 1.11).

In this chapter, we discuss the differential equations of motion for irro-tational, inviscid, adiabatic and shock-free motion of a perfect gas.


Crocco’s Theorem

Consider two-dimensional, steady, inviscid flow in natural coordinates(l , n) such that l is along the streamline direction and n is perpendicularto the direction of the streamline. The advantage of using natural coor-dinate system – a coordinate system in which one coordinate is alongthe streamline direction and other normal to it – is that the flow veloc-ity is always along the streamline direction and the velocity normal tostreamline is zero.


Though this is a two-dimensional flow, we can apply one-dimensionalanalysis, by considering the portion between the two streamlines 1 and2 (as shown in Figure 5.1) as a streamtube and taking the third dimen-sion to be ∞.

l

2

1 ∆n

V

R

p + ∆p

p

n

Figure 5.1Flow between two streamlines.


Let us consider unit width in the third direction, for the present study.For this flow, the equation of continuity is

ρ V ∆n = constant (5.1)

The l-momentum equation is 1

ρ V ∆n dV = −dp ∆n

1Momentum equation. For incompressible flow,

X

Fi = ρ

Z

Vx dQ

where Q is the volume flow rate. For compressible flow,

X

Fi =

Z

ρVx dQX

dFi = ρVx dQ = mVx


The l-momentum equation can also be expressed as

ρ V∂V∂l

= −∂p∂l

(5.2)

The n-momentum equation is

dV = 0

But there will be centrifugal force acting in the n-direction. Therefore,

ρ V 2

R= −∂p

∂n(5.3)


The energy equation is

h +V 2

2= h0 (5.4)

Also, by Eq. (1.55),

T ds = dh − dpρ

Differentiation of Eq. (5.4) gives dh+VdV = dh0. Therefore, the entropyequation becomes

T ds = −(

V dV +dpρ

)

+ dh0


This equation can be split as follows.

(i)

T∂s∂l

= −(

V∂V∂l

+1ρ

∂p∂l

)

becausedh0

dl= 0 along the streamlines.

(ii)

T∂s∂n

= −(

V∂V∂n

+1ρ

∂p∂n

)

+dh0

dn


Introducing ∂p/∂l from Eq. (5.2) and ∂p/∂n from Eq. (5.3) into theabove two equations, we get

T∂s∂l

= 0 (5.5a)

T∂s∂n

= −V(

∂V∂n

− VR

)

+dh0

dn(5.5b)

But(V

R − ∂V∂n

)

= ζ is the vorticity of the flow. Therefore,

T∂s∂n

=dh0

dn+ V ζ (5.6)

This is known as Crocco’s theorem for two-dimensional flows. From thisit is seen that the rotation depends on the rate of change of entropy andstagnation enthalpy normal to the streamlines.


Crocco’s theorem essentially relates entropy gradients to vorticity, insteady, frictionless, non-conducting, adiabatic flows. In this form Crocco’sequation shows that if entropy (s) is a constant, the vorticity (ζ) must bezero. Likewise, if vorticity ζ is zero, the entropy gradient in the direc-tion normal to the streamline (ds/dn) must be zero, implying that theentropy s is a constant. That is, isentropic flows are irrotational andirrotational flows are isentropic. This result is true, in general, only forsteady flows of inviscid fluids in which there are no body forces actingand the stagnation enthalpy is a constant.

From Eq. (5.5a) it is seen that the entropy does not change along astreamline. Also, Eq. (5.5b) shows how entropy varies normal to thestreamlines.


The circulation is

Γ =

∮

cV dl =

∫ ∫

scurl V ds =

∫ ∫

sζ ds (5.7)

By Stokes theorem, the velocity ζ is given by

ζ = curl V (5.8)

ζx =

(

∂Vz

∂y− ∂Vy

∂z

)

ζy =

(

∂Vx

∂z− ∂Vz

∂x

)

ζz =

(

∂Vy

∂x− ∂Vx

∂y

)

where ζx , ζy , ζz are the vorticity components.


The two conditions that are necessary for a frictionless flow to be isen-tropic throughout are

1 h0 = constant, throughout the flow.2 ζ = 0, throughout the flow.

From Eq. (5.8), ζ = 0 for irrotational flow. That is, if a frictionless flow isto be isentropic, the total enthalpy should be constant throughout andthe flow should be irrotational.


When ζ 6= 0

Since h0 = constant, T0 = constant (perfect gas). For this type of flowwe can show that,

ζ =TV

dsdn

= −R T0

V p0

dp0

dn(5.9)

From Eq. (5.9), it is seen that in an irrotational flow (i.e. with ζ = 0),stagnation pressure does not change normal to the streamlines. Evenwhen there is a shock in the flow field, p0 changes along the streamlinesat the shock, but does not change normal to the streamlines.


Let h0 = constant (isoenergic flow). Then Eq. (5.6) can be written invector form as

T grad s + V × curl V = grad h0 (5.10a)

where grad s stands for increase of entropy s in the n-direction. For asteady, inviscid and isoenergic flow,

T grad s + V × curl V = 0

V × curl V = −T grad s (5.10b)

If s = constant, V × curl V = 0. This implies that (a) the flow isirrotational, i.e. curl V = 0, or (b) V is parallel to curl V .


Irrotational flow

For irrotational flows (curl V = 0), a potential function φ exists such that,

V = grad φ (5.11)

Therefore, the velocity components are given by

Vx =∂φ

∂x, Vy =

∂φ

∂y, Vz =

∂φ

∂z

The advantage of introducing φ is that the three unknowns Vx , Vy andVz in a general three-dimensional flow are reduced to a singleunknown φ. With φ, the irrotationality conditions defined by Eq. (5.8)may be expressed as follows.


ζx =∂Vz

∂y− ∂Vy

∂z= 0 =

∂

∂y

(

∂φ

∂z

)

− ∂

∂z

(

∂φ

∂y

)

= 0

Also, the incompressible continuity equation ▽.V = 0 becomes

∂2φ

∂x2 +∂2φ

∂y2 +∂2φ

∂z2 = 0

This is Laplace’s equation. With the introduction of φ, the three equa-tions of motion can be replaced, at least for incompressible flow, by oneLaplace’s equation, which is a linear equation.


Basic Solutions of Laplace’s Equation

We know from our basic studies on fluid flows (Fluid Mechanics - AnIntroduction, E. Rathakrishnan, 2007) that,

1 For uniform flow (towards positive x-direction), the potential func-tion is

φ = V∞ x

2 For a source of strength Q, the potential function is

φ =Q2π

ln r


1 For a doublet of strength µ (issuing in negative x-direction), thepotential function is

φ =µ cos θ

r2 For a potential (free) vortex (counterclockwise) with circulation Γ,

the potential function is

φ =Γ

2πθ


The General Potential Equation for Three-Dimensional Flow

For a steady, inviscid, three-dimensional flow, by continuity equation,

▽ . (ρ V ) = 0

i.e.∂(ρ Vx )

∂x+

∂(ρ Vy )

∂y+

∂(ρ Vz)

∂z= 0 (5.12)


Euler’s equations of motion (neglecting body forces) are:

ρ

Vx∂Vx

∂x+ Vy

∂Vx

∂y+ Vz

∂Vx

∂z

= −∂p∂x

(5.13a)

ρ

Vx∂Vy

∂x+ Vy

∂Vy

∂y+ Vz

∂Vy

∂z

= −∂p∂y

(5.13b)

ρ

Vx∂Vz

∂x+ Vy

∂Vz

∂y+ Vz

∂Vz

∂z

= −∂p∂z

(5.13c)


For incompressible flows, the density ρ is a constant. Therefore, theabove four equations are sufficient for solving the four unknowns Vx ,Vy , Vz and p. But for compressible flows, ρ is also an unknown. There-fore, the unknowns are ρ, Vx , Vy , Vz and p. Hence, the additionalequation, namely, the isentropic process equation, is used. That is,p/ργ = constant is the additional equation used along with continuityand momentum equations.


Introducing the potential function φ, we have the velocity componentsas

Vx =∂φ

∂x= φx , Vy =

∂φ

∂y= φy , Vz =

∂φ

∂z= φz (5.14)

Equation (5.12) may also be written as

ρ

∂Vx

∂x+

∂Vy

∂y+

∂Vz

∂z

+ Vx∂ρ

∂x+ Vy

∂ρ

∂y+ Vz

∂ρ

∂z= 0 (5.12a)


From isentropic process relation, ρ = ρ(p). Hence,

∂ρ

∂x=

dρ

dp∂p∂x

= − 1a2 ρ

Vx∂Vx

∂x+ Vy

∂Vx

∂y+ Vz

∂Vx

∂z

because from Eq. (5.13a),

∂p∂x

= −ρ

Vx∂Vx

∂x+ Vy

∂Vx

∂y+ Vz

∂Vx

∂z

,dpdρ

= a2


Similarly,

∂ρ

∂y= − 1

a2 ρ

Vx∂Vy

∂x+ Vy

∂Vy

∂y+ Vz

∂Vy

∂z

∂ρ

∂z= − 1

a2 ρ

Vx∂Vz

∂x+ Vy

∂Vz

∂y+ Vz

∂Vz

∂z

With the above relations for∂ρ

∂x,

∂ρ

∂yand

∂ρ

∂z, Eq. (5.12a) can be ex-

pressed as


∂Vx

∂x

1 − V 2

x

a2

+

∂Vy

∂y

1 −V 2

y

a2

+∂Vz

∂z

1 − V 2

z

a2

− Vx Vy

a2

∂Vx

∂y+

∂Vy

∂x

−Vy Vz

a2

∂Vy

∂z+

∂Vz

∂y

− Vz Vx

a2

∂Vz

∂x+

∂Vx

∂z

= 0


But the velocity components and their derivatives in terms of potentialfunction are the following.

Vx =∂φ

∂x= φx , Vy =

∂φ

∂y= φy , Vz =

∂φ

∂z= φz

∂Vx

∂x= φxx ,

∂Vy

∂y= φyy ,

∂Vz

∂z= φzz

∂Vx

∂y= φxy ,

∂Vy

∂z= φyz ,

∂Vz

∂x= φzx


Therefore, in terms of potential function φ, the above equation can beexpressed as

1 − φ2

x

a2

φxx +

1 −φ2

y

a2

φyy +

1 − φ2

z

a2

φzz

−2

φx φy

a2 φxy +φy φz

a2 φyz +φz φx

a2 φzx

= 0 (5.15)

This is the basic potential equation for compressible flow and it is non-linear.


The difficulties associated with compressible flow stem from the fact thatthe basic equation is nonlinear. Hence the superposition of solutions isnot valid. Further, in Eq. (5.15) the local speed of sound ‘a’ is also avariable. By Eq. (2.9e), we have

aa∞

2= 1 − γ − 1

2M2

∞

V 2x + V 2

y + V 2z

V 2∞

− 1

(5.16)

To solve a compressible flow problem, we have to solve Eq. (5.15) usingEq. (5.16), but this is not possible analytically. However, numericalsolution is possible for given boundary conditions.


Linearization of the Potential Equation

The general equation for compressible flows, namely Eq. (5.15), canbe simplified for flow past slender or planar bodies. Aerofoil, slenderbodies of revolution and so on are typical examples for slender bodies.Bodies like wing, where one dimension is smaller than others, are calledplanar bodies. These bodies introduce small disturbances. The aerofoilcontour becomes the stagnation streamline.

For the aerofoil shown in Figure 5.2, with the exception of nose region,the perturbation velocity w is small everywhere.


Small Perturbation Theory

Assume that the velocity at any point in the flow field is given by thevector sum of the freestream velocity V∞ along the x-axis and the per-turbation velocity components u, v and w along x , y and z-directions,respectively. Consider the flow around an aerofoil shown in Figure 5.2.

x

V

w

Vx

Vz w

V∞

z

Figure 5.2Aerofoil in an uniform flow.


The velocity components around the aerofoil are

Vx = V∞ + u, Vy = v , Vz = w (5.17)

where Vx , Vy , Vz are the main flow velocity components and u, v , ware the perturbation (disturbance) velocity components along x , y andz directions, respectively.

The small perturbation theory postulates that the perturbation velocitiesare small compared to the main velocity components, i.e.

u ≪ V∞, v ≪ V∞, w ≪ V∞ (5.18a)


Therefore,Vx ≈ V∞, Vy ≪ V∞, Vz ≪ V∞ (5.18b)

Now, consider a flow at small angle of attack or yaw as shown in Figure5.3. Here,

Vx = V∞ cos α + u, Vy = V∞ sin α + v

V∞

α

Figure 5.3Aerofoil at an angle of attack in an uniform flow.


Since the angle of attack α is small, the above equations reduce to

Vx = V∞ + u, Vy = v

Thus, Eq. (5.17) can be used for this case also.With Eq. (5.17), linearization of Eq. (5.15) gives

(1 − M2)φxx + φyy + φzz = 0 (5.19)

neglecting all higher order terms, where M is the local Mach number.Therefore, Eq. (5.16) should be used in solving Eq. (5.19).


The perturbation velocities may also be written in potential form, asfollows:

Let φ = φ∞ + ϕ, where

φ∞ = V∞x : φxx = ր0φ∞xx + ϕxx . . .

Therefore, φ may be called the disturbance (perturbation) potential andhence the perturbation velocity components are given by

u =∂φ

∂x, v =

∂φ

∂y, w =

∂φ

∂z(5.20)


With the assumptions of small perturbation theory, Eq. (5.16) can beexpressed as

aa∞

2= 1 − (γ − 1) M2

∞

uV∞

(5.21)

a∞

a

2=

1 − (γ − 1) M2∞

uV∞

−1

Using Binomial theorem, (a∞/a)2 can be expressed as

a∞

a

2= 1 + (γ − 1)

uV∞

M2∞

+ O

M4

∞

u2

V 2∞

(5.22)


Substituting the above expression for (a∞/a) in the equation

M =

1 +u

V∞

a∞

a

M∞

the relation between the local Mach number M and freestream Machnumber M∞ may be expressed as (neglecting small terms)

M2 =

[

1 + 2u

V∞

1 +γ − 1

2M2

∞

]

M2∞

(5.23)


The combination of Eqs. (5.23) and (5.19) gives

(1 − M2∞

)φxx + φyy + φzz =2

V∞

M2∞

φx φxx

1 +γ − 1

2M2

∞

(5.24)

Equation (5.24) is a nonlinear equation and is valid for subsonic, tran-sonic and supersonic flow under the framework of small perturbationswith u ≪ V∞, v ≪ V∞ and w ≪ V∞. It is, however, not valid for hyper-sonic flow even for slender bodies (since u ≈ V∞ in the hypersonic flowregime). The equation is called the linearized potential flow equation,though it is not linear.


Equation (5.24) may also be written as

(1−M2∞

)φxx +φyy +φzz = 2M2

∞

1 − M2∞

uV∞

1 +γ − 1

2M2

∞

(1−M2∞

)φxx

(5.25)Further linearization is possible if

M2∞

1 − M2∞

uV∞

≪ 1 (5.26)

With this condition Eq. (5.25) results in

(1 − M2∞

)φxx + φyy + φzz = 0 (5.27)

This is the fundamental equation governing most of the compressibleflow regime.


Equation (5.27) is valid only when Eq. (5.26) is valid and Eq. (5.26) isvalid only when the freestream Mach number M∞ is sufficiently differentfrom 1. Hence, Eq. (5.26) is valid for subsonic and supersonic flowsonly. For transonic flows, Eq. (5.24) can be used. For M∞ ≈ 1, Eq.(5.24) reduces to

−γ + 1V∞

φx φxx + φyy + φzz = 0 (5.28)

The nonlinearity of Eq. (5.28) makes transonic flow problems muchmore difficult than subsonic or supersonic flow problems.

Equation (5.27) is elliptic (i.e. all terms are positive) for M∞ < 1 andhyperbolic (i.e. not all terms are positive) for M∞ > 1. But in both thecases, the governing differential equation is linear. This is the advan-tage of Eq. (5.27).


Potential Equation for Bodies of Revolution

Fuselage of airplane, rocket shells, missile bodies and circular ducts arethe few bodies of revolutions which are commonly used in practice. Thegeneral three-dimensional Cartesian equations can be used for theseproblems. But it is much simpler to use cylindrical polar coordinatesthan Cartesian coordinates. Cartesian coordinates are x , y , z and thecorresponding velocity components are Vx , Vy , Vz . The cylindrical polarcoordinates are x , r , θ and the corresponding velocity components areVx , Vr , Vθ.


For axisymmetric flows with cylindrical coordinates, the equations willbe independent of θ. Thus, mathematically, cylindrical coordinates re-duce the problem to two dimensional. However, for flows which are notaxially symmetric (e.g. missile at an angle of attack), θ will be involved.The continuity equation in cylindrical coordinates is

∂(ρ Vx )

∂x+

1r

∂(ρ r Vr )

∂r+

1r

∂(ρ Vθ)

∂θ= 0 (5.29)


Expressing the velocity components in terms of the potential function φas

Vx =∂φ

∂x, Vr =

∂φ

∂r, Vθ =

1r

∂φ

∂θ(5.30)

The potential Eq. (5.25) can be written, in cylindrical polar coordinates,as


1 − φ2

x

a2

φxx +

1 − φ2

r

a2

φrr +

1 − 1

r2

φ2θ

a2

1r2 φθθ

−2

φx φr

a2 φxr +φx φθ

a2

1r2 φxθ +

φr φθ

a2

1r2 φrθ

+

1 +

1r2

φ2θ

a2

1r

φr = 0

(5.31)


Also,

aa∞

2= 1 − γ − 1

2M2

∞

V 2x + V 2

r + V 2θ

V 2∞

− 1

(5.32)

The small perturbation assumptions are:

Vx = V∞ + u, Vr = vr , Vθ = vθ

u ≪ V∞, vr ≪ V∞, vθ ≪ V∞

where Vx , Vr , Vθ are the mean velocity components and u, vr , vθ arethe perturbation velocity components along the x-, r - and θ-direction,respectively.


Introduction of these relations in Eq. (5.31) results in

(1 − M2)φxx + φrr +1r

φr +1r2 φθθ = 0 (5.33)

where M is the local Mach number after Eq. (5.23). The relations for u,vr , vθ in polar coordinates, under small perturbation assumption are

u =∂φ

∂x= φx , vr =

∂φ

∂r= φr , vθ =

1r

∂φ

∂θ=

1r

φθ


With these expressions for u, vr and vθ, Eq. (5.24) can be written as

(1 − M2∞

)φxx + φrr +1r

φr +1r2 φθθ =

2V∞

M2∞

φx φxx

1 +γ − 1

2M2

∞

(5.34)This equation corresponds to Eq. (5.24) with the same term on the righthand side. Therefore, with

M2∞

1 − M2∞

uV∞

≪ 1


Equation (5.34) simplifies to

(1 − M2∞

)φxx + φrr +1r

φr +1r2 φθθ = 0 (5.35)

This is the governing equation for subsonic and supersonic flows incylindrical coordinates. For transonic flow, Eq. (5.35) becomes

−γ + 1V∞

φx φxx + φrr +1r

φr +1r2 φθθ = 0 (5.36)

For axially symmetric subsonic and supersonic flows, φθθ = 0. There-fore, Eq. (5.35) reduces to

(1 − M2∞

)φxx + φrr +1r

φr = 0 (5.37)


Similarly, Eq. (5.36) reduces to

−γ + 1V∞

φx φxx + φrr +1r

φr = 0 (5.38)

Equation (5.38) is the equation for axially symmetric transonic flows.All these equations are valid only for small perturbations, i.e. for smallvalues of angle of attack and angle of yaw (< 15◦).


Conclusions

From the above discussions on potential flow theory for compressibleflows, we can draw the following conclusions.

1 The small perturbation equations for subsonic and supersonic flowsare linear, but for transonic flows the equation is nonlinear.

2 Subsonic and supersonic flow equations do not contain the specificheats ratio γ, but transonic flow equation contains γ. This showsthat the results obtained for subsonic and supersonic flows, withsmall perturbation equations, can be applied to any gas, but thiscannot be done for transonic flows.


1 All these equations are valid for slender bodies. This is true ofrockets, missiles, etc.

2 These equations can also be applied to aerofoils, but not to bluffshapes like circular cylinder, etc.

3 For non-slender bodies, the flow can be calculated by using theoriginal nonlinear equation.


Solution of Nonlinear Potential Equation

(i) Numerical methods:

The nonlinearity of Eq. (5.24) makes it tedious to solve the equationanalytically. However, solution for the equation can be obtained by nu-merical methods. But a numerical solution is not a general solution, andis valid only for a specific configuration of flow field with a fixed Machnumber and specified geometry.


(ii) Transformation (Hodograph) methods:

When one velocity component is plotted against another velocity com-ponent, the resulting curve may be linear, whereas in the physical plane,the relation may be nonlinear. This method is used for solving certaintransonic flow problems.

(iii) Similarity methods:

In these methods, the boundary conditions need to be specified for solv-ing the equation. Detailed discussion of this method can be found inChapter 6 on Similarity Methods.


Examine the streamlines around an aerofoil kept in a flow field as shownin Figure 5.4.

V∞

x

z

f(x, y, z)

Figure 5.4Cambered aerofoil at an angle of attack.


In inviscid flow the streamline near the boundary is similar to the bodycontour. The flow must satisfy the following boundary conditions (BCs):

Boundary condition 1 – Kinetic flow condition.

The flow velocity at all surface points are tangential on the body contour.The component of velocity normal to the body contour is zero.


Boundary condition 2

At z → ±∞, perturbation velocities are zero or finite. The kinematic flowcondition for the aerofoil shown in Figure 5.4, with small perturbationassumptions, may be written as follows.

Body contour: f = f (x , y , z)

The velocity vector V at any point on the body is tangential to the sur-face. Therefore, on the surface of the aerofoil, (V . grad f ) = 0, i.e.

(V∞ + u)∂f∂x

+ v∂f∂y

+ w∂f∂z

= 0 (5.39)


But u/V∞ ≪ 1, therefore, Eq. (5.39) simplifies to

V∞

∂f∂x

+ v∂f∂y

+ w∂f∂z

= 0 (5.40)

For two-dimensional flows, v = 0; ∂f/∂y = 0. Therefore, Eq. (5.39)reduces to

wV∞ + u

= −∂f/∂x∂f/∂z

=

∂z∂x

c(5.41)

where the subscript ‘c’ refers to the body contour and (∂z/∂x) is theslope of the body and u and v are the tangential and normalcomponents of velocity, respectively. Expressing u and w as powerseries of z, we get


u(x , z) = u(x , 0) + a1z + a2z2 + . . .

w(x , z) = w(x , 0) + b1z + b2z2 + . . .

The coefficients a′s and b′s in these series are functions of x . If thebody is sufficiently slender,

w(x , 0)

V∞ + u(x , 0)=

dzdx

c

i.e. for sufficiently slender bodies, it is not necessary to fulfill the bound-ary condition on the contour of the aerofoil. It is sufficient if the bound-ary condition on the x−axis of the body is satisfied, i.e. on the axis ofa body of revolution or the chord of an aerofoil. With u/V∞ ≪ 1, theabove condition becomes

w(x , 0)

V∞

=

dzdx

c(5.42)


For planar bodies: ∂f/∂y = 0 and therefore,

w(x , y , 0)

V∞

=

dzdx

c(5.43)

i.e. the condition is satisfied in the plane of the body. In Eqs. (5.42) and(5.43), the elevation of the body above the x−axis is neglected.


Bodies of Revolution

For bodies of revolution, the term 1r

∂∂r (rvr ) present in the continuity

equation (5.29) becomes finite. Because of this term the perturbationsnear the body become significant. Therefore, a power series for velocitycomponents is not possible. However, we can apply the following ap-proximation to express the perturbation velocity as a power series. Foraxisymmetric bodies,

1r

∂

∂r(rvr ) ∼

∂u∂x

,∂

∂r(rvr ) ∼ r

∂u∂x

when r → 0; ∂∂r (rvr ) ≈ 0 or rvr = a0(x).


Thus, even though the radial component of velocity vr on the axis of abody of revolution is of the order of 1/r , it can be estimated near the axissimilar to a potential vortex. For a potential vortex, the radial velocity is

vr ∝1r

Now, vr can be expressed in terms of a power series as

rvr = a0 + a1r + a2r2 + . . .


For the axisymmetric body with its surface profile contour given by thefunction R(x), we have

vr

V∞ + u=

dR(x)

dx

c

The simplified kinematic flow condition for the body in Figure 5.5 is

(rvr )0

V∞

= R(x)dR(x)

dx(5.44)

where subscript 0 refers to the axis of the body.


x

r

V∞

R(x)

Figure 5.5An axisymmetric body in a flow.


Equation (5.44) is called the simplified kinematic flow condition in thesense that the kinematic flow condition is fulfilled on the axis, ratherthan on the surface of the body contour.

On the axis of the body, Eq. (5.44) gives

limr→0

(rvr ) = V∞ R(x)dR(x)

dx(5.45)

From the above discussions, it may be summarized that the boundaryconditions for this kind of problem are the following.


For two-dimensional (planar) bodies,

wV∞ + u

c≈ (w)0

V∞

=

∂z∂x

c(5.46)

For bodies of revolution (elongated bodies),

R(x)

vr

V∞ + u

c≈ (rvr )0

V∞

= R(x)dR(x)

dx(5.47)


Pressure Coefficient

Pressure coefficient is the non-dimensional difference between a localpressure and the freestream pressure. The idea of finding the velocitydistribution is to find the pressure distribution and then integrate it toget lift, moment, and pressure drag. For three-dimensional flows, thepressure coefficientCp is given by (Eq. 2.54) is

Cp =2

γ M2∞

{

[

γ − 12

M2∞

1 − (V∞ + u)2 + v2 + w2

V 2∞

+ 1

]γ/(γ−1)

− 1

}


or

Cp =2

γ M2∞

{

[

1 − γ − 12

M2∞

2uV∞

+u2 + v2 + w2

V 2∞

]γ/(γ−1)

− 1

}

where M∞ and V∞ are the freestream Mach number and velocity, re-spectively, u, v and w are the x , y and z-components of perturbationvelocity and γ is the ratio of specific heats. Expanding the RHS of thisequation binomially and neglecting the third and higher-order terms ofthe perturbation velocity components, we get

Cp = −

2

uV∞

+ (1 − M2∞

)u2

V 2∞

+u2 + w2

V 2∞

(5.48)


For two-dimensional or planar bodies, the Cp simplifies further, resultingin

Cp = −2u

V∞

(5.48a)

This is a fundamental equation applicable to three-dimensional com-pressible (subsonic and supersonic) flows, as well as for low speedtwo-dimensional flows.


Bodies of Revolution

For bodies of revolution, by small perturbation assumption, we haveu ≪ V∞, but v and w are not negligible. Therefore, Eq. (5.48) simplifiesto

Cp = −2u

V∞

− v2 + w2

V 2∞

(5.49)

The above equation, which is in Cartesian coordinates, may also beexpressed as

Cp = −2u

V∞

−

vr

V∞

2(5.50)


Combining Eqs. (5.47) and (5.50), we get

Cp = −2u

V∞

−

dR(x)

dx

2

(5.51)

where R(x) is the expression for the body contour.


Summary

In this chapter we have presented some of the aspects of the linearizedcompressible flow. It is important to recognize the fundamental natureof the approximations introduced to linearize the basic potential equa-tion for compressible flow. Although modern numerical techniques arecapable of yielding accurate solutions for flows with complex geome-tries, linearized solutions still play a vital role in the field of compressibleflows.


By Crocco’s theorem, we have, for two-dimensional flows, the equation

T∂s∂n

=dh0

dn+ V ζ

where s is entropy, h0 is stagnation enthalpy and ζ is vorticity. Thisequation relates the velocity of a flow field to the entropy of the fluid.Also, it stipulates the conditions under which a frictionless flow will havethe same entropy on different streamlines, that is, it will be isentropic.


The conditions are

h0 = constant, throughout the flow

ζ = 0, throughout the flow

That is, isentropic flows are irrotational and irrotational flows are isen-tropic. This result is true, in general, only for steady flows of inviscidfluids in which there are no body forces and in which the stagnationenthalpy is constant.


For irrotational flows, by Laplace’s equation, we have

∂2φ

∂x2 +∂2φ

∂y2 +∂2φ

∂z2 = 0

The basic solutions for this equations are the following.

1 For uniform flow (towards positive x-direction), the potential func-tion is

φ = V∞ x

2 For a source of strength Q, the potential function is

φ =Q2π

ln r


1 For a doublet of strength µ (issuing in negative x-direction), thepotential function is

φ =µ cos θ

r2 For a potential (free) vortex (counterclockwise) with circulation Γ,

the potential function is

φ =Γ

2πθ


The governing equations for three-dimensional potential flow in Carte-sian coordinates are the following.

The continuity equation

∂(ρ Vx )

∂x+

∂(ρ Vy )

∂y+

∂(ρ Vz)

∂z= 0

The Euler’s equation of motion (neglecting body forces) or the momen-tum equations are


ρ

Vx∂Vx

∂x+ Vy

∂Vx

∂y+ Vz

∂Vx

∂z

= −∂p∂x

ρ

Vx∂Vy

∂x+ Vy

∂Vy

∂y+ Vz

∂Vy

∂z

= −∂p∂y

ρ

Vx∂Vz

∂x+ Vy

∂Vz

∂y+ Vz

∂Vz

∂z

= −∂p∂z


and the isentropic process relation is

pp0

=

ρ

ρ0

γ

Instead of the energy equation, we have the isentropic relation. Theisentropic relation given above is for a perfect gas. The more generalform of the isentropic relation is, simply, s = constant.


In terms of the velocity potential, the momentum equation becomes

1 − φ2

x

a2

φxx +

1 −φ2

y

a2

φyy +

1 − φ2

z

a2

φzz

−2

φx φy

a2 φxy +φy φz

a2 φyz +φz φx

a2 φzx

= 0

This is the basic potential equation for compressible flow and it is non-linear. Because of the nonlinearity of this equation, the superpositionof solutions is not valid. Furthermore, the local speed of sound in thisequation is also a variable.


With small perturbation theory, Eq. (5.15) reduces to

(1 − M2∞

)φxx + φyy + φzz = 0

This is an approximate equation; it no longer represents the exact physicsof the flow. However, the original nonlinear equation [Eq. (5.15)] hasbeen reduced to a linear equation [Eq. (5.19)]. It is also called the lin-earized perturbation velocity potential equation. It is important to notethat, Eq. (5.19) is valid for subsonic and supersonic flows only. It is notvalid for transonic flows.


The linearized potential flow equation valid for subsonic, supersonic andtransonic flows is

(1 − M2∞

)φxx + φyy + φzz =2

V∞

M2∞

φx φxx

1 +γ − 1

2M2

∞

Equation (5.24) is nonlinear unlike Eq. (5.19) which is linear.

From the above governing equations it is evident that subsonic and su-personic flows lend themselves to approximate, linearized theory forthe case of irrotational, isentropic flow with small perturbations. Buttransonic and hypersonic flows cannot be linearized, even with smallperturbations.


The potential equations for bodies of revolution in cylindrical polar coor-dinates are the following.

The continuity equation

∂(ρ Vx )

∂x+

1r

∂(ρ r Vr )

∂r+

1r

∂(ρ Vθ)

∂θ= 0


The momentum equation

1 − φ2

x

a2

φxx +

1 − φ2

r

a2

φrr +

1 − 1

r2

φ2θ

a2

1r2 φθθ

−2

φx φr

a2 φxr +φx φθ

a2

1r2 φxθ +

φr φθ

a2

1r2 φrθ

+

1 +

1r2

φ2θ

a2

1r

φr = 0


With small perturbation assumption, this equation reduces to

(1 − M2∞

)φxx + φrr +1r

φr +1r2 φθθ = 0

This equation is valid for subsonic and supersonic flows in cylindricalcoordinates. For transonic flows the governing equation is

−γ + 1V∞

φx φxx + φrr +1r

φr +1r2 φθθ = 0


For axially symmetric flows, φθθ = 0; therefore, Eqs. (5.35) and (5.36)reduce to

(1 − M2∞

)φxx + φrr +1r

φr = 0

−γ + 1V∞

φx φxx + φrr +1r

φr = 0

All these equations are valid only for small perturbations, that is, forsmall values of angle of attack and yaw (< 15◦). The pressure coeffi-cient Cp given by small perturbation theory is

Cp = −

2

uV∞

+ (1 − M2∞

)u2

V 2∞

+u2 + w2

V 2∞


For bodies of revolution, the Cp becomes

Cp = −2u

V∞

−

dR(x)

dx

2

where R(x) is the expression for body contour.In the light of the discussion of this chapter it may be summarized that,


The small perturbation equations for subsonic and supersonic flowsare linear, but for transonic flows the equation is nonlinear.

Subsonic and supersonic equations do not contain the specificheats ratio γ, but transonic flow equation contains γ. This impliesthat the results obtained for subsonic and supersonic flows withsmall perturbation equations can be applied to any gas, but thiscannot be done for transonic flows.


Exercise Problems

1 Show that for compressible flow of a perfect gas, the variation of totalpressure across a streamline is given by

− 1ρ0

dp0

dn=

1 +γ − 1

2M2

uζ +γ − 1

2cp M2 dT0

dn

where n is the direction normal to the streamline.


2 The nose of a cylindrical body has the profile R = εx3/2, 0 ≤ x ≤ 1.Show that, the pressure distribution on the body is given by

Cp

ε2 = 6x ln2

ε√

M2 − 1− 3x ln(x) − 33

4x

Estimate the drag coefficient for M =√

2 and ε = 0.1.

(Hint: For obtaining CD, use CDS(L) =∫ L

0 Cp(x) S′(x)dx , where S(x)is the cross-sectional area of the body at x and L is the length of thebody).

[Ans. CD = 0.0786]


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Applied Gas Dynamics Compressible Flow Equations - … · Applied Gas Dynamics Compressible Flow Equations Ethirajan Rathakrishnan Applied Gas Dynamics, John Wiley & Sons (Asia) Pte