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Next: Subsonic Flow Past a Up: Two-Dimensional Compressible Inviscid Flow Previous: Homenergic Homentropic Flow

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

$\displaystyle u$ $\displaystyle = U,$ (15.102)
$\displaystyle v$ $\displaystyle =0.$ (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

$\displaystyle u$ $\displaystyle = U + u',$ (15.104)
$\displaystyle v$ $\displaystyle = v',$ (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

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

It then follows from Equation (15.100) that

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

$\displaystyle (1-{\rm Ma}_\infty^{\,2})\,\frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y}$ $\displaystyle \simeq {\rm Ma}_\infty^{\,2}\,(\gamma+1)\,\frac{u'}{U}\,\frac{\pa...
...{\rm Ma}_\infty^{\,2}\,(\gamma-1)\,\frac{u'}{U}\,\frac{\partial v'}{\partial y}$    
  $\displaystyle \phantom{=} + {\rm Ma}_\infty^{\,2}\,\frac{v'}{U}\left(\frac{\partial u'}{\partial y}+\frac{\partial v'}{\partial x}\right).$ (15.108)

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

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

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

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

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

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

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

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

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

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

Hence, we obtain

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

which reduces to

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

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

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


$\displaystyle f(x,y)=0$ (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,

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

which reduces to

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

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

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

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

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


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

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

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

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

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

$\displaystyle (1-{\rm Ma}_\infty^{\,2})\,\frac{\partial^{\,2} \phi}{\partial x^{\,2}} + \frac{\partial^{\,2}\phi}{\partial y^{\,2}}$ $\displaystyle \simeq 0,$ (15.127)
$\displaystyle C_p$ $\displaystyle \simeq \frac{2}{U}\,\frac{\partial\phi}{\partial x},$ (15.128)
$\displaystyle \frac{\partial\phi(x,0)}{\partial y}$ $\displaystyle \simeq -U\left(\frac{dy}{dx}\right)_{\rm body},$ (15.129)


next up previous
Next: Subsonic Flow Past a Up: Two-Dimensional Compressible Inviscid Flow Previous: Homenergic Homentropic Flow
Richard Fitzpatrick 2016-03-31