Small-Perturbation Theory

A great number of problems of interest in compressible fluid mechanics are concerned with the perturbation of a known flow pattern. The most common case is that of uniform, steady flow. Let $U$ denote the uniform flow velocity, which is directed parallel to the $x$ -axis. The density, pressure, and temperature are also assumed to be uniform, and are denoted $\rho_\infty$ , $p_\infty$ , and $T_\infty$ , respectively. The corresponding sound speed is $c_\infty$ , and the Mach number is ${\rm Ma}_\infty= U/c_\infty$ . Finally. the velocity field of the unperturbed flow pattern is

$$u = U,$$ (15.102)

$$v =0.$$ (15.103)

Suppose that a solid body, such as an airfoil, is placed in the aforementioned flow pattern. The cross-section of the body is assumed to be independent of the Cartesian coordinate $z$ . The body disturbs the flow pattern, and changes its velocity field, which is now written

$$u = U + u',$$ (15.104)

$$v = v',$$ (15.105)

where $u'$ and $v'$ are known as induced velocity components. We are interested in situations in which $u'/U\ll 1$ and $v'/U\ll 1$ .

Equation (15.101) can be combined with the previous two equations to give

$$c^{\,2} = c_\infty^{\,2}-\frac{1}{2}\,(\gamma-1)\left(2\,U\,u'+u'^{\,2}+v'^{\,2}\right).$$ (15.106)

It then follows from Equation (15.100) that

$$(1-{\rm Ma}_\infty^{\,2})\,\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y} = {\rm Ma}_\infty^{\,2}\left[(\gamma+1)\,\frac{u'}{U} +\left(\fra... ...ma-1}{2}\right)\left(\frac{v'}{U}\right)^2\right]\frac{\partial u'}{\partial x}$$

$$\phantom{=}+ {\rm Ma}_\infty^{\,2}\left[(\gamma-1)\,\frac{u'}{U} ... ...ma-1}{2}\right)\left(\frac{u'}{U}\right)^2\right]\frac{\partial v'}{\partial y}$$

$$\phantom{=} + {\rm Ma}_\infty^{\,2}\,\frac{v'}{U}\left(1+\frac{u'... ...ght)\left(\frac{\partial u'}{\partial y}+\frac{\partial v'}{\partial x}\right).$$ (15.107)

The previous equation is exact. However, if $u'/U$ and $v'/V$ are small then it becomes possible to neglect many of the terms on the right-hand side. For instance, neglecting terms that are third-order in small quantities, we obtain

$$(1-{\rm Ma}_\infty^{\,2})\,\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y} \simeq {\rm Ma}_\infty^{\,2}\,(\gamma+1)\,\frac{u'}{U}\,\frac{\pa... ...{\rm Ma}_\infty^{\,2}\,(\gamma-1)\,\frac{u'}{U}\,\frac{\partial v'}{\partial y}$$

$$\phantom{=} + {\rm Ma}_\infty^{\,2}\,\frac{v'}{U}\left(\frac{\partial u'}{\partial y}+\frac{\partial v'}{\partial x}\right).$$ (15.108)

Furthermore, if we neglect terms that are second-order in small quantities then we get the linear equation

$$(1-{\rm Ma}_\infty^{\,2})\,\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y} \simeq 0.$$ (15.109)

Note, however, than in so-called transonic flow, where ${\rm Ma}_\infty \simeq 1$ , the coefficient of $\partial u'/\partial x$ on the left-hand side of Equation (15.108) becomes very small. In this situation, it is not possible to neglect the first term on the right-hand side. However, the condition ${\rm Ma}_\infty \simeq 1$ does not affect the term $\partial v'/\partial y$ on the left-hand side of Equation (15.108), and so the other terms on the right-hand side can still be neglected. Thus, transonic flow is governed by the non-linear equation

$$(1-{\rm Ma}_\infty^{\,2})\,\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y} \simeq {\rm Ma}_\infty^{\,2}\,(\gamma+1)\,\frac{u'}{U}\,\frac{\partial u'}{\partial x}.$$ (15.110)

On the other hand, subsonic (i.e., ${\rm Ma}_\infty < 1$ ) and supersonic flow (i.e., ${\rm Ma}_\infty > 1$ ) are both governed by Equation (15.109). Another situation in which certain terms on the right-hand side of Equation (15.108) must be retained is hypersonic flow (i.e., ${\rm Ma}_\infty \gg 1$ ). This follows because, although $u'/U$ and $v'/V$ are small, their products with ${\rm Ma}_\infty^{\,2}$ can still be non-negligible. Roughly speaking, Equation (15.109) is valid for $0\leq {\rm Ma}_\infty \leq 0.8$ and $1.2\leq {\rm Ma}_\infty \leq 5$ . In other words, transonic flow corresponds to $0.8<{\rm Ma}_\infty < 1.2$ , and hypersonic flow to ${\rm Ma}_\infty>5$ (Anderson 2003).

The pressure coefficient is defined

$$C_p = \frac{p-p_\infty}{(1/2)\,\rho_\infty\,U^{\,2}}=\frac{2}{\gamma\,{\rm Ma}_\infty^{\,2}}\,\frac{p-p_\infty}{p_\infty}.$$ (15.111)

(See Section 15.9.) Equation (15.101) implies that

$$c^{\,2} + \frac{1}{2}\,(\gamma-1)\,q^{\,2} = c_\infty^{\,2}+\frac{1}{2}\,(\gamma-1)\,U^{\,2},$$ (15.112)

where $q^{\,2}= (U+u')^{\,2} + v'^{\,2}$ . Moreover, Equation (14.61) yields

$$\frac{p}{p_\infty} = \left[\frac{2+(\gamma-1)\,{\rm Ma}_\infty^{\,2}}{2+(\gamma-1)\,{\rm Ma}^{\,2}}\right]^{\,\gamma/(\gamma-1)},$$ (15.113)

where ${\rm Ma}=q/c$ . The previous two equations can be combined to give

$$\frac{p}{p_\infty} = \left[1+\frac{1}{2}\,(\gamma-1)\,{\rm Ma}_\infty^{\,2}\left(1-\frac{q^{\,2}}{U^{\,2}}\right)\right]^{\,\gamma/(\gamma-1)}.$$ (15.114)

Hence, we obtain

$$C_p = \frac{2}{\gamma\,{\rm Ma}_\infty^{\,2}}\left(\left[1+\frac{... ...2}\left(1-\frac{q^{\,2}}{U^{\,2}}\right)\right]^{\,\gamma/(\gamma-1)}-1\right),$$ (15.115)

which reduces to

$$C_p = \frac{2}{\gamma\,{\rm Ma}_\infty^{\,2}}\left(\left[1-\frac{... ...\frac{u'^{\,2}+v'^{\,2}}{U^{\,2}}\right)\right]^{\,\gamma/(\gamma-1)}-1\right).$$ (15.116)

Using the binomial expansion on the expression in square brackets, and neglecting terms that are third-order, or higher, in small quantities, we obtain

$$C_p \simeq -\left[\frac{2\,u'}{U}+(1-{\rm Ma}_\infty^{\,2})\left(\frac{u'}{U}\right)^{\,2}+ \left(\frac{v'}{U}\right)^2\right].$$ (15.117)

For two-dimensional flows, in the limit in which Equation (15.109) is valid, it is consistent to retain only first-order terms in the previous equation, so that

$$C_p\simeq -\frac{2\,u'}{U}.$$ (15.118)

Let

$$f(x,y)=0$$ (15.119)

be the equation of the surface of the solid body that perturbs the flow. At the surface, the velocity vector of the flow must be perpendicular to the local normal: that is, the flow must be tangential to the surface. In other words,

$${\bf q}\cdot\nabla f = 0,$$ (15.120)

which reduces to

$$(U+u')\,\frac{\partial f}{\partial x} + v'\,\frac{\partial f}{\partial y} =0.$$ (15.121)

Neglecting $u'$ with respect to $U$ , we obtain

$$\frac{v'}{U} \simeq -\frac{\partial f/\partial x}{\partial f/\partial y} = \frac{dy}{dx},$$ (15.122)

where $dy/dx$ is the slope of the surface, and $v'/U$ the approximate slope of a streamline.

Now, the body has to be thin in order to satisfy our assumption that the induced velocities are relatively small. This implies that the coordinate $y$ differs little from zero (say) on the surface of the body. Hence, we can write

$$v'(x,y) = v'(x,0) + \left(\frac{\partial v'}{\partial y}\right)_{y=0} y+\cdots$$ (15.123)

in the immediate vicinity of the surface. Within the framework of small-perturbation theory, it is consistent to neglect all terms on the right-hand side of the previous equation after the first. Hence, the boundary condition (15.122) reduces to

$$v'(x,0)= U\left(\frac{dy}{dx}\right)_{\rm body},$$ (15.124)

respectively.

Because a homenergic, homentropic flow pattern is necessarily irrotational (see Section 15.10), we can write

$${\bf q} = U\,{\bf e}_x -\nabla\phi,$$ (15.125)

where $\phi$ is the perturbed velocity potential. (See Section 4.15.) It follows that

$$u' = - \frac{\partial\phi}{\partial x}, v' =-\frac{\partial\phi}{\partial y}.$$ (15.126)

Hence, Equations (15.109), (15.118), and (15.124) become

$$(1-{\rm Ma}_\infty^{\,2})\,\frac{\partial^{\,2} \phi}{\partial x^{\,2}} + \frac{\partial^{\,2}\phi}{\partial y^{\,2}} \simeq 0,$$ (15.127)

$$C_p \simeq \frac{2}{U}\,\frac{\partial\phi}{\partial x},$$ (15.128)

$$\frac{\partial\phi(x,0)}{\partial y} \simeq -U\left(\frac{dy}{dx}\right)_{\rm body},$$ (15.129)

respectively.