Unsteady Aerodynamics with Case Studies

Abstract

Three-dimensional unsteady aerodynamics and computational approaches for continuation of Chap. 8 in unsteady Aerodynamics are elaborated in the present chapter (Reproduced from and based on several papers: [1,2,3,4]). Further it illustrates some of the contributions in lifting surfaces based on potential aerodynamic approaches and serves to provide and elaborate a good spectrum of problems and analytical approaches focusing on case studies in three-dimensional unsteady aerodynamic theory and computational methods. The first case study illustrates one of the first attempts in unsteady lifting potential flow solution method using geometric discretization and singularity distributions on the aerodynamic surface, which contributes to the present Boundary Element Method. The second case study addresses a more complex situation, that is the unsteady subsonic three-dimensional flow with separation bubble, and utilizes the singularity method. The third case study reflects the application of these previous techniques in addressing aircraft buffeting phenomena, a practical aeroelastic problem, using dynamic response approach The last case study elaborates the application of the three-dimensional unsteady aerodynamics lifting surface method to address combined aeroelasticity and acoustic excitation in a unified boundary element approach. These case studies are presented to provide an illustration of the application of lifting surface or boundary element method to solve unsteady aerodynamic problems.

Appendix: For Case Study II—Calculation of 3D Unsteady Subsonic Flow with Separation Bubble Using Singularity Method

1.1 The Singular Part of Kernel Functions

Noting the complexity of the kernel functions, it is intended to integrate them using numerical approach. It is obvious that the singularity contained in the kernel functions will tend to spoil the numerical integration if they are not carefully treated. The usual technique to take care of this problem is to isolate the region in the vicinity of the singular point, and to integrate it in an analytical manner applying a suitable limiting process. The remaining regular kernel can then be easily and accurately integrated using a standard numerical quadrature. The limiting processes for the second and third kernels are well documented in many literatures. Such procedure leads to Eqs. (11.98) and (11.99). Afterward, the singularities of the first and fourth kernel are presented.

The integral in the chordwise direction will contain singularities if η and η ̓take the same values. This situation is closely related to the integration technique applied for the spanwise direction. The singular part of the first kernel can be examined to consist of some poles and a logarithmic function, since the Bessel function has a logarithmic behaviour for small arguments. This singular part is [42]

$$K_{{1s}} \left( {\xi ,\eta ;\xi ^{'} ,\eta ^{'} } \right) = \frac{{e^{{ - {\text{i}}k\left( {\xi - \xi ^{'} } \right)}} }}{{s^{2} }}\left( \begin{gathered} - \frac{1}{{\left( {\eta - \eta ^{'} } \right)^{2} }}\left| {1 + \frac{{\xi - \xi ^{'} }}{{r^{'} }}} \right| \hfill \\ + \frac{{ik}}{{r^{'} }} - \frac{{k^{2} }}{2}\ln \left| {r^{'} - \left( {\xi - \xi ^{'} } \right)} \right| \hfill \\ \end{gathered} \right)$$

(11.191)

The singular part of the fourth kernel has a less complexity since the singularity is caused only by the r ̓. Consequently it produces some poles, which can be taken separately as

$$ K_{1s} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) \approx \frac{1}{2\pi }\left[ { - \frac{ik}{{\beta^{2} \left| {\xi - \xi^{^{\prime}} } \right|}}\left( {1 - \frac{{M\left( {\xi - \xi^{^{\prime}} } \right)}}{{\left| {\xi - \xi^{^{\prime}} } \right|}}} \right)} \right] $$

(11.192)

If these two singular parts obtained are subtracted from the original kernels, the rest will become regular functions which facilitate the numerical treatment of them. Proceeding along this line, then the non-singular of the first and fourth kernel are

$$ K_{1ns} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) = K_{1} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) - K_{1s} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) $$

(11.193)

$$ K_{4ns} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) = K_{4} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) - K_{4s} \left( {\xi ,\eta ;\xi^{^{\prime}} ,\eta^{^{\prime}} } \right) $$

(11.194)

1.2 Integration Along the Chordwise Direction

The integration of the first kernel is worked out exactly in the same manner as the one in Laschka [42], where the singular parts are treated analytically and the rest numerically. Proceeding along Laschka’s approach the singular integral is divided into three parts¹⁵

$$\begin{gathered} f_{{1r}} \left( {\xi ,\eta ;\xi ^{'} ,\eta ^{'} } \right) = \frac{1}{{2\pi }}\sum\limits_{{r = 0}}^{R} {\left\{ {\mathop {\lim }\limits_{{\varepsilon \to 0}} \left[ \begin{gathered} \frac{2}{\varepsilon }a_{r}^{*} \left( \eta \right)f_{{11r\,s}} \left( {\xi ,\eta ;\eta ^{'} } \right) \hfill \\ - \int\limits_{\begin{subarray}{l} - 1 \\ \text{singular} \\ {\text{integral}} \end{subarray} }^{1} {\frac{{a*_{r} \left( {\eta ^{'} } \right)}}{{\left( {\eta - \eta ^{'} } \right)}}} f_{{11r\,\,ns}} \left( {\xi ,\eta ;\eta ^{'} } \right)d\eta ^{'} \hfill \\ \end{gathered} \right]} \right\}} \\ + \int\limits_{{ - 1}}^{1} {a_{r}^{*} } \left( {\eta ^{'} } \right)\ln \left| {\eta - \eta ^{'} } \right|f_{{12r}} \left( {\xi ,\eta ;\eta ^{'} } \right)d\eta ^{'} + \int\limits_{{ - 1}}^{1} {a_{r}^{*} } \left( {\eta ^{'} } \right)f_{{13r}} \left( {\xi ,\eta ;\eta ^{'} } \right)d\eta ^{'} \\ \end{gathered}$$

(11.195)

The Gauss-Jacobian Quadrature is employed for the integration of the non-singular kernel times the series expansion. The quadrature chosen takes the advantage of the series expansion, h_r(X), as the weighting function,

$$ L\left( {r,X} \right) = \int\limits_{0}^{1} {h_{r} } \left( {X^{^{\prime}} } \right)F\left( {X,\eta ;X^{^{\prime}} ,\eta^{^{\prime}} } \right)dX^{^{\prime}} = \sum\limits_{j = 0}^{J} {B_{rj} F} \left( {X,\eta ;X_{j} ,\eta^{^{\prime}} } \right) $$

(11.196)

where

$$ B_{rJ} = \frac{{\cos \left( {r\phi_{j} } \right)\cos \left( {\left( {r + 1} \right)\phi_{j} } \right)}}{J + 1} $$

(11.197)

The components of $f_{1r} \left( {\xi ,\eta ;\eta } \right)$ are then evaluated one by one using this formula as

$$ f_{11rs} \left( {\xi ,\eta ;\eta^{^{\prime}} } \right) = \sum\limits_{j = 0}^{J} {B_{rj} } e^{{ - {\text{i}}k\frac{c\left( \eta \right)}{s}\left( {X - X{}_{j}} \right)}} \left( {1 + \frac{{X - X{}_{j}}}{{R_{j} }}} \right) $$

(11.198)

$$\begin{aligned} f_{{12r\nu s}} \left( {\xi ,\eta ;\eta ^{'} } \right) & = f_{{1rs}} \left( {\xi ,\eta ;\eta ^{'} } \right) \\ & + \left( {\frac{{dh_{r} \left( X \right)}}{{dX^{'} }} + ik\frac{{c\left( \eta \right)}}{s}h_{r} \left( {X^{'} } \right)} \right) \\ & \quad \left( {\frac{{\beta s}}{{c\left( \eta \right)}}} \right)^{2} \left( {\eta - \eta ^{'} } \right)^{2} \ln \left| {\eta - \eta ^{'} } \right| \\ \end{aligned}$$

(11.199)

$$\begin{aligned} f_{{12r}} \left( {\xi ,\eta ;\eta ^{'} } \right) & = - k^{2} \sum\limits_{{j = 0}}^{J} {B_{{rj}} } e^{{ - {\text{i}}k\frac{{c\left( \eta \right)}}{s}\left( {X - X{}_{j}} \right)}} \\ & \quad + \left( {\frac{{\beta s}}{{c\left( \eta \right)}}} \right)^{2} \left[ {\frac{{dh_{r} \left( X \right)}}{{dX^{'} }} + ik\frac{{c\left( \eta \right)}}{s}h_{r} \left( X \right)} \right] - 2ik\frac{s}{{c\left( {\eta ^{'} } \right)}}h_{r} \left( X \right) \\ \end{aligned}$$

(11.200)

and

$$ \begin{aligned} f_{13r} \left( {\xi ,\eta ;\eta^{^{\prime}} } \right) & = \frac{{f_{1r\nu s} \left( {\xi ,\eta ;\eta^{^{\prime}} } \right)}}{{\left( {\eta - \eta^{^{\prime}} } \right)^{2} }} - \ln \left| {\eta - \eta^{^{\prime}} } \right|f_{2r} \left( {\xi ,\eta ;\eta^{^{\prime}} } \right) \\ & - \frac{1}{{\left( {\eta - \eta^{^{\prime}} } \right)^{2} }}\sum\limits_{j = 0}^{J} {B_{rj} } e^{{ - {\text{i}}k\frac{c\left( \eta \right)}{s}\left( {X - X{}_{j}} \right)}} \\ & \left\{ \begin{gathered} k\left| {\eta - \eta^{^{\prime}} } \right|\left[ \begin{gathered} K_{1} \left( {k\left| {\eta - \eta^{^{\prime}} } \right|} \right) - i + \frac{\pi i}{2}\left( {I_{1} \left( {k\left| {\eta - \eta^{^{\prime}} } \right|} \right) - L_{1} k\left| {\eta - \eta^{^{\prime}} } \right|} \right) \hfill \\ + i\int\limits_{0}^{U \cdot Br} {\frac{\tau }{{\left( {1 + \tau^{2} } \right)^{\frac{1}{2}} }}} e^{{{\text{2i}}k\left( {\eta - \eta ^{\prime}} \right)\tau }} d\tau \hfill \\ \end{gathered} \right] \hfill \\ + \frac{{X - X_{j} }}{R}e^{{{\text{ - i}}k\frac{c\left( \eta \right)}{{s\beta^{2} }}\left( {X - X_{j} - MR_{j} } \right)}} \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

(11.201)

where the upper boundary of the integral becomes

$$ U \cdot Br = \frac{{\left( {X - X^{^{\prime}} - MR} \right)}}{{\beta^{2} \left| {\eta - \eta^{^{\prime}} } \right|}}\frac{{c\left( {\eta^{^{\prime}} } \right)}}{s} $$

(11.202)

and

$$ R = \left[ {\left( {X - X^{^{\prime}} } \right)^{2} + \beta^{2} \frac{{s^{2} }}{{c^{2} \left( \eta \right)}}\left( {\eta - \eta^{^{\prime}} } \right)^{2} } \right]^{\frac{1}{2}} $$

(11.203)

$$ X^{^{\prime}} = \frac{{X^{^{\prime}} - X_{{{\text{LE}}}} }}{c\left( \eta \right)} = \frac{1}{2}\left( {1 - \cos \left( {\phi^{^{\prime}} } \right)} \right) $$

(11.204)

$$ \phi^{^{\prime}} = \frac{j\pi }{{J + 1}} $$

(11.205)

$$ j = 0,1,2, \ldots j $$

(11.206)

Following Watkins et al.[54], the approximation of the integrand is carried out by expansion like,

$$ \frac{\tau }{{\left( {1 + \tau^{2} } \right)^{\frac{1}{2}} }} = 1 - 1.101e^{ - 0.329\tau } - 0.899e^{ - 1.4067\tau } - 0.09480933e^{ - 0.329\tau } \sin \pi \tau $$

(11.207)

Using this expression the integral appeared in Eq. (11.201) could be worked out.