The Linearized Velocity Potential Equation

Consider the two-dimensional, irrotational, isentropic flow over the body shown in Figure 11.2. The body is immersed in a uniform flow with velocity Voo oriented in the x direction. At an arbitrary point P in the flow field, the velocity is V with the x and у components given by и and v, respectively. Let us now visualize the velocity V as the sum of the uniform flow velocity plus some extra increments in velocity. For example, the x component of velocity и in Figure 11.2 can be visualized as Voo plus an increment in velocity (positive or negative). Similarly, the у component of velocity v can be visualized as a simple increment itself, because the uniform flow has a zero component in the у direction. These increments are called perturbations, and

и — Voo + й v = v

where u and v are called the perturbation velocities. These perturbation velocities are not necessarily small; indeed, they can be quite large in the stagnation region in front of the blunt nose of the body shown in Figure 11.2. In the same vein, because V = Уф, we can define a perturbation velocity potential ф such that

ф = Voo* + ф

дф

dx

дф

dy

Hence,

З 2ф д2ф

дх ду дх ду

Substituting the above definitions into Equation (11.12), and multiplying by a2, we obtain

[1 1.14]

Equation (11.14) is called the perturbation velocity potential equation. It is precisely the same equation as Equation (11.12) except that it is expressed in terms of ф instead of ф. It is still a nonlinear equation.

To obtain better physical insight in some of our subsequent discussion, let us recast Equation (11.14) in terms of the perturbation velocities. From the definition of ф given earlier, Equation (11.14) can be written as

r9 /Т7 Л, 7. 9 „лЗе лл3л

[a ~ (Voo + и)2]— + (a2 – v2)—– 2(Тоо + u)v— = 0

dx dy dy

From the energy equation in the form of Equation (8.32), we have

Кэс _ I (Too + n)2 + P2

2 “ у – 1 +

Substituting Equation (11.15) into (11.14a), and algebraically rearranging, we obtain

й у + 1 й2 у — їй2

tr + ,)^ + — vi + ~K

‘ й у + 1 v2

2 V^ +

Equation (11.16) is still exact for irrotational, isentropic flow. Note that the left-hand side of Equation (11.16) is linear but the right-hand side is nonlinear. Also, keep in mind that the size of the perturbations й and v can be large or small; Equation (11.16) holds for both cases.

Let us now limit our considerations to small perturbations; that is, assume that the body in Figure 11.2 is a slender body at small angle of attack. In such a case, й and v will be small in comparison with V-^. Therefore, we have

V2 ’ V2

Keep in mind that products of й and v with their derivatives are also very small. With this in mind, examine Equation (11.16). Compare like terms (coefficients of like derivatives) on the left – and right-hand sides of Equation (11.16). We find

1. For 0 < Moo £ 0.8 or Moo > 1.2, the magnitude of

or in terms of the perturbation velocity potential,

Examine Equation (11.18). It is a linear partial differential equation, and therefore is inherently simpler to solve than its parent equation, Equation (11.16). However, we have paid a price for this simplicity. Equation (11.18) is no longer exact. It is only an approximation to the physics of the flow. Due to the assumptions made in obtaining Equation (11.18), it is reasonably valid (but not exact) for the following combined situations:

1. Small perturbation, that is, thin bodies at small angles of attack

2. Subsonic and supersonic Mach numbers

In contrast, Equation (11.18) is not valid for thick bodies and for large angles of attack. Moreover, it cannot be used for transonic flow, where 0.8 < Мж < 1.2, or for hypersonic flow, where Мж > 5.

We are interested in solving Equation (11.18) in order to obtain the pressure distribution along the surface of a slender body. Since we are now dealing with approximate equations, it is consistent to obtain a linearized expression for the pres­sure coefficient—an expression which is approximate to the same degree as Equation (11.18), but which is extremely simple and convenient to use. The linearized pressure coefficient can be derived as follows.

First, recall the definition of the pressure coefficient Cp given in Section 1.5:

C„ = -—— [11.19]

QoC

where qoo = 5Pco^tx; = dynamic pressure. The dynamic pressure can be expressed in terms of M0c as follows:

Substituting Equation (11.21) into (11.19), we have

[11.22]

Equation (11.22) is simply an alternate form of the pressure coefficient expressed in terms of Moo. It is still an exact representation of the definition of C

To obtain a linearized form of the pressure coefficient, recall that we are dealing with an adiabatic flow of a calorically perfect gas; hence, from Equation (8.39),

V2 V2

T+- = roo + -^ [11.23]

2,Cp 2cp

Recalling from Equation (7.9) that cp = yR/(y — 1), Equation (11.23) can be written as

T – Tso = —— 25———–

°° 2yR/(y – 1)

Also, recalling that = *Jy Equation (11.24) becomes T x у – 1 – У 2 ^ у _ ! y2 _ V2

Too 2 yRToo 2 a^

In terms of the perturbation velocities

V2 = (Voo + m)2 + v2

Equation (11.25) can be written as

Since the flow is isentropic, p/poo = (T/TooyRy 1 ’, and Equation (11.26) becomes

2 ‘ 00 V V2 ‘ [2]oo ‘oo

Equation (11.27) is still an exact expression. However, let us now make the as­sumption that the perturbations are small, that is, й/V.^ 1, u2/ <SY 1, and

v2/ <gc 1. In this case, Equation (11.27) is of the form

— = (1 – e)^"1) [11.28]

Poo

where e is small. From the binomial expansion, neglecting higher-order terms, Equa­tion (11.28) becomes

p v

— = 1——– -—£ + ••• [11.29]

Poo Y – 1

Comparing Equation (11.27) to (11.29), we can express Equation (11.27) as

7- = 1 –

Poo 2

Substituting Equation (11.30) into the expression for the pressure coefficient, Equa­tion (11.22), we obtain

	cP =	2	_i – ^	Ґ 2й | й2 + v2 ^	■ – 1
	умі	VEoo+ ) +
or	cP =	2 й Й2 + v2			[1 1.31]

Since й2/and v2/V^ <ж. 1, Equation (11.31) becomes

Equation (11.32) is the linearized form for the pressure coefficient; it is valid only for small perturbations. Equation (11.32) is consistent with the linearized perturbation velocity potential equation, Equation (11.18). Note the simplicity of Equation (11.32); it depends only on the x component of the velocity perturbation, namely, й.

To round out our discussion on the basics of the linearized equations, we note that any solution to Equation (11.18) must satisfy the usual boundary conditions at infinity and at the body surface. At infinity, clearly ф = constant; that is, й = v — 0. At the body, the flow-tangency condition holds. Let в be the angle between the tangent to the surface and the freestream. Then, at the surface, the boundary condition is obtained from Equation (3.48c):

v v

tan# = – = ———– – [11.33]

и Ех, + и

which is an exact expression for the flow-tangency condition at the body surface. A simpler, approximate expression for Equation (11.33), consistent with linearized theory, can be obtained by noting that for small perturbations, й « Hence, Equation (11.33) becomes v

Equation (11.34) is an approximate expression for the flow-tangency condition at the body surface, with accuracy of the same order as Equations (11.18) and (11.32).