Far-Field Asymptotics and Zonal Structure of Theoretical Flow Models

Abstract

This chapter starts from the far-field behaviours of velocity field in externally-unbounded flow. We find that the well-known algebraic decay of disturbance velocity as derived kinematically is too conservative. Once the kinetics is taken into account by working on the fundamental solutions of far-field linearized Navier-Stokes equations, it is proven that the furthest far-field zone adjacent to the uniform fluid at infinity must be unsteady, viscous and compressible, where all disturbances degenerate to sound waves that decay exponentially. But this optimal rate does not exist in some commonly used simplified flow models, such as steady flow, incompressible flow and inviscid flow, because they actually work in true subspaces of the unbounded free space, which are surrounded by further far fields of different nature. This finding naturally leads to a zonal structure of externally-unbounded flow field. The significance of the zonal structure is demonstrated by its close relevance to existing theories of aerodynamic force and moment in external flows, including the removal of the difficulties or paradoxes inherent in the simplified models.

Appendix: A General Theorem About Fundamental Solution

For self-contained, a general theorem about fundamental solution first given by Lagerstrom et al. [10] is repeated here. This theorem will be applied to various cases in this chapter and Chap. 3.

The theorem is derived for solutions defined in an n-dimensional vector space $\mathbf {R}^n$ whose points are denoted by ${\varvec{x}},{\varvec{\xi }}$. Let $\mathbf {M}_1$ and $\mathbf {M}_2$ be two linear differential matrix operators defined on vector function over $\mathbf {R}^n$. Then the problem is the determination of the fundamental matrix (tensor) $\mathbf {G}({\varvec{x}},{\varvec{\xi }})$ for the differential equation

$$\begin{aligned} (a\mathbf {M}_1-b\mathbf {M}_2-k^2\mathbf {I}) \cdot {\varvec{u}}= - {\varvec{f}}({\varvec{x}}), \end{aligned}$$

(2.5.1)

where a, b, k are constants, ${\varvec{f}}({\varvec{x}})$ is a given vector function defined over $\mathbf {R}^n$ and vanishing suitably at infinity. The solution ${\varvec{u}}({\varvec{x}})$ to (2.5.1) is also a vector function and the fundamental solution satisfying homogeneous boundary conditions is

(2.5.2)

Linear Differential System Theorem. If $\mathbf {M}_1$ and $\mathbf {M}_2$ are linear differential matrix operators such that

$$\begin{aligned} \mathbf {M}_1 \cdot \mathbf {M}_2 = \mathbf {M}_2 \cdot \mathbf {M}_1 = \mathbf 0, \end{aligned}$$

(2.5.3a)

and

$$\begin{aligned} \mathbf {M}_1 - \mathbf {M}_2 = L\mathbf{{I}}, \end{aligned}$$

(2.5.3b)

where

$$\begin{aligned} \mathbf{{I}}= & {} n\text {-dimensional unit matrix}, \end{aligned}$$

(2.5.3c)

$$\begin{aligned} L= & {} \text {a scalar linear differential operator}, \end{aligned}$$

(2.5.3d)

then the fundamental solution $\mathbf {G}({\varvec{x}},{\varvec{\xi }})$ of (2.5.1) is given by

$$\begin{aligned} \mathbf {G}({\varvec{x}},{\varvec{\xi }}) = \frac{1}{k^2} \left( \mathbf {M}_1 g_{\sqrt{\frac{k^2}{a}}} - \mathbf {M}_2 g_{\sqrt{\frac{k^2}{b}}} \right) , \end{aligned}$$

(2.5.4)

where $g_k({\varvec{x}},{\varvec{\xi }})$ is the fundamental solution of the scalar differential equation formed with L,

$$\begin{aligned} (L-k^2) {\varvec{v}}= -{\varvec{f}}({\varvec{x}}). \end{aligned}$$

(2.5.5)

Here $g_k({\varvec{x}},{\varvec{\xi }})$ is defined, in the usual way, by the requirement that ${\varvec{v}}({\mathbf {x}})$ satisfying (2.5.5) and homogenous boundary conditions be given by

$$\begin{aligned} {\varvec{v}}({\varvec{x}}) = \int _{\mathbf {R}^n} g_k({\varvec{x}},{\varvec{\xi }}) {\varvec{f}}({\varvec{\xi }}) \mathrm {d} {\varvec{\xi }}. \end{aligned}$$

(2.5.6)

The theorem will now be proved. In the proof it is convenient to use the fact that

$$\begin{aligned} L g_0 = 0 \end{aligned}$$

(2.5.7)

in order to represent $\mathbf {G}$ in another form. We may write

$$\begin{aligned} \mathbf {G}({\varvec{x}},{\varvec{\xi }}) = \frac{1}{a}\mathbf {M}_1 U_1 - \frac{1}{b} \mathbf {M}_2 U_2, \end{aligned}$$

(2.5.8a)

when we define

$$\begin{aligned} U_1 = \frac{a}{k^2} \left( g_{\sqrt{\frac{k^2}{a}}} - g_0 \right) , \quad U_2 = \frac{b}{k^2} \left( g_{\sqrt{\frac{k^2}{b}}} - g_0 \right) . \end{aligned}$$

(2.5.8b)

This is equivalent to (2.5.4). $U_1$ and $U_2$ are now regular at ${\varvec{x}}={\varvec{\xi }}$ because the singularity of $g_k$ is independent of k.

Proof

The theorem will be proved if it is shown that

$$\begin{aligned} (a\mathbf {M}_1-b\mathbf {M}_2-k^2\mathbf {I}) \cdot \int _{\mathbf {R}^n} \mathbf {G}({\varvec{x}},{\varvec{\xi }}) \cdot {\varvec{f}}({\varvec{\xi }}) \mathrm {d} {\varvec{\xi }}= - {\varvec{f}}({\varvec{x}}), \end{aligned}$$

(2.5.9a)

which is equivalent to (2.5.2). Using the second form of $\mathbf {G}$ (2.5.8) we may transform the equation (2.5.9a) to be proved by taking the operators $\mathbf {M}_1$ and $\mathbf {M}_2$ in $\mathbf {G}$ outside the integral sign. This is allowed because $U_1$ and $U_2$ are regular for all ${\varvec{x}}$. Thus

$$\begin{aligned} {\begin{matrix} &{} \left( \mathbf {M}_1^2 - \frac{k^2}{a}\mathbf {M}_1\right) \cdot \int _{\mathbf {R}^n} U_1({\varvec{x}},{\varvec{\xi }}) {\varvec{f}}({\varvec{\xi }}) \mathrm {d} {\varvec{\xi }}\\ -&{} \left( -\mathbf {M}_2^2 - \frac{k^2}{b}\mathbf {M}_2\right) \cdot \int _{\mathbf {R}^n} U_2({\varvec{x}},{\varvec{\xi }}) {\varvec{f}}({\varvec{\xi }}) \mathrm {d} {\varvec{\xi }}= -{\varvec{f}}, \end{matrix}} \end{aligned}$$

(2.5.9b)

where we have used (2.5.3). It also follows from (2.5.3) that

$$\begin{aligned} \mathbf {M}_1^2 = L\mathbf {M}_1, \quad \mathbf {M}_2^2 = -L\mathbf {M}_2, \end{aligned}$$

so that the left hand side of (2.5.9b) may be transformed as follows:

$$\begin{aligned}&\mathbf {M}_1 \left( L-\frac{k^2}{a} \right) \cdot \int _{\mathbf {R}^n} U_1 {\varvec{f}}\mathrm {d} {\varvec{\xi }}- \mathbf {M}_2 \left( L-\frac{k^2}{b} \right) \cdot \int _{\mathbf {R}^n} U_2 {\varvec{f}}\mathrm {d} {\varvec{\xi }}\\= & {} \frac{a}{k^2}\mathbf {M}_1 \cdot \left[ \left( L-\frac{k^2}{a} \right) \int _{\mathbf {R}^n} g_{\sqrt{ \frac{k^2}{a} } } {\varvec{f}}\mathrm {d} {\varvec{\xi }}- L \int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}\right] + \mathbf {M}_1 \cdot \int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}\\&-\frac{b}{k^2}\mathbf {M}_2 \cdot \left[ \left( L-\frac{k^2}{b} \right) \int _{\mathbf {R}^n} g_{\sqrt{ \frac{k^2}{b} } } {\varvec{f}}\mathrm {d} {\varvec{\xi }}- L \int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}\right] - \mathbf {M}_2 \cdot \int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}\\= & {} \frac{a}{k^2}\mathbf {M}_1 \cdot (-{\varvec{f}}+ {\varvec{f}}) - \frac{b}{k^2}\mathbf {M}_2 \cdot (-{\varvec{f}}+ {\varvec{f}}) + (\mathbf {M}_1 - \mathbf {M}_2) \cdot \int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}. \end{aligned}$$

Thus

$$\begin{aligned} (\mathbf {M}_1 - \mathbf {M}_2) \cdot \int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}\equiv L\int _{\mathbf {R}^n} g_{0} {\varvec{f}}\mathrm {d} {\varvec{\xi }}= -{\varvec{f}}, \end{aligned}$$

(2.5.10)

which proves (2.5.9a) and hence the theorem.

$\square$