Stability of Vortex Symmetry at Flow Separation from Slender Bodies and Control by Local Gas Heating

Abstract

A new approach to describe the asymmetry vortex state occurrence for the separated flow over slender bodies is presented. On the basis of the proposed model, a criterion of the asymmetry origin for conical bodies is found using catastrophe theory. Main properties of the transition to an asymmetric state are studied on the basis of the local analysis, the flow characteristics near the critical saddle point. Using the obtained criterion and the new model, numerical calculations of turbulent boundary layer are made to estimate an effectiveness of global flow structure control methods using local plasma discharge or surface heating. The qualitative confirmation of presented numerical results was done by experiments.

1. Introduction

The symmetric vortex structure stability problem for the separated flow over slender bodies has important practical applications, and it is intensively investigated using numerical and experimental methods during a long enough period. Reviews of these investigations including theoretical, experimental and numerical investigations are presented in works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. However, reasons and mechanism of a spontaneous transition to asymmetry are not defined completely up to date. It is known that this phenomenon can be initiated by different flow heterogeneities, for example, small surface deformations [5, 6], gas blow-suction through holes [7], acoustic waves and other incoming flow disturbances.

Experiments and calculations show that even very small disturbances, such as technological surface defects, a numerical grid imperfection and rounding computer errors, can lead to the flow asymmetry. Also, outer reasons of the flow asymmetry arising exist, which are connected with the laminar-turbulent transition [8] or the vortex destruction [9], but they are not considered in this work.

There are two different points of view on asymmetry arising: the convective amplification of stationary disturbances, generated near the tip [7, 11], or an absolute disturbances instability, related with the flow velocity profile in the saddle point [4, 5, 12]. However, none of these approaches allows explaining experimental observations. Also, numerical modeling results do not allow understanding the asymmetry reason [13, 14, 15, 16]. Previous theoretical investigations based on the model vortex-cut also do not allow revealing the transition mechanism to asymmetric state [17, 18, 19].

It is assumed in the first part of the present work that the arising asymmetry is an inner property of the symmetric flow and is related with its structure (global) instability. The problem is reduced to the stationary (critical) point analysis of the nonlinear dynamic system, which is described by symmetric flow streamline equations. Strong mathematical results were obtained only for autonomic gradient systems, which correspond to conical flows. In this case, on the basis of the catastrophe theory [20], a symmetric flow stability criterion is found, which is confirmed by experimental data [4, 21, 22, 23]. On the basis of nonlinear local equation properties near the critical point, the qualitative analysis of characteristics for the transition to asymmetric state is done.

The obtained asymmetry origin criterion allowed proposing a method for the global flow structure control based on the local gas heating by the plasma discharge [1, 2, 3, 21, 22, 23]. The new model for the description of the plasma discharge in the boundary layer was proposed, and on this basis, calculations of the separation point location for the thin round cone are presented and verified using experimental data [21, 22, 23].

2. Problem formulation

Let consider the separated stationary flow around slender-pointed body (Figure 1a and b) with affine-like transverse section form $Y=F\left(Z\right)B\left(X\right)$, where $B\left(0\right)=0$ corresponds to the body nose and $X,Y,Z$ are nondimensional Descartes coordinates:

Figure 1.

Flow scheme. (a) General view. (b) Transverse section. 1 is the vortex sheet, and 2 is the separation location.

$$X=\frac{X^{\ast }}{l},Y=\frac{Y^{\ast }}{\mathit{\delta l}},Z=\frac{Z^{\ast }}{\mathit{\delta l}},r=\frac{r^{\ast }}{\mathit{\delta l}}.$$

Here, l is the body length, and $a^{\ast }=\mathit{\delta l}$ is its thickness. It is assumed that the flow and the body are symmetrical with respect to the plane $Z=0$_.

It is assumed that, for the Reynolds number Re, the angle of attack ${\alpha }^{\ast }$_, and the nondimensional body thickness $\delta$, the following relations are fulfilled:

$$\delta \ll 1,\delta Re=\delta \frac{U_{\infty }l}{{\nu }_{\infty }}\gg 1,\alpha =\frac{{\alpha }^{\ast }}{\delta }=O\left(1\right).$$E1

Here, $U_{\infty }$ is the incoming flow speed, and ${\nu }_{\infty }$ is the kinematic viscosity.

At conditions (1), solutions of Navier-Stokes equations may be found using asymptotic expansions of flow parameters at $\delta \to 0$ and $Re\to \infty$ [24]:

$$u^{\ast }=U_{\infty }\left[1+{\delta }^{2}ln\delta u_0\left(X\right)+{\delta }^{2}u\left(X,Y,Z\right)+\dots \right],$$

$$w^{\ast }=U_{\infty }\mathit{\delta w}\left(X,Y,Z\right)+\dots ,v^{\ast }=U_{\infty }\mathit{\delta v}\left(X,Y,Z\right)+\dots ,$$

$$p^{\ast }-p_{\infty }={\rho }_{\infty }{U}_{\infty }^2{\delta }^{2}p\left(X,Y,Z\right)+\dots ,{\rho }^{\ast }={\rho }_{\infty }\left[1+{\delta }^2\rho \left(X,Y,Z\right)+\dots \right].$$E2

Here, $u^{\ast }$,$v^{\ast }$, and $w^{\ast }$ are velocity components along axes$X,Y,Z$; $p^{\ast }$ and ${\rho }^{\ast }$ are pressure and density, and $p_{\infty }$ and ${\rho }_{\infty }$are its non-perturbed values.

In the first-order approximation, the regular limit of Navier-Stokes equations for the asymptotic expansion (2) is the Euler equation system for the incompressible fluid in the crossflow plane with the non-flow boundary condition:

$$\frac{\partial v}{\partial X}+v\frac{\partial v}{\partial Y}+w\frac{\partial v}{\partial Z}+\frac{\partial p}{\partial Y}=0,\frac{\partial w}{\partial X}+v\frac{\partial w}{\partial Y}+w\frac{\partial w}{\partial Z}+\frac{\partial p}{\partial Z}=0,\frac{\partial v}{\partial Y}+\frac{\partial w}{\partial Z}=0;{\left[v-\mathit{wB}\frac{\mathit{dF}}{\mathit{dZ}}\right]}_{Y=\mathit{BF}}=F\frac{\mathit{dB}}{\mathit{dX}}$$E3

In these equations, the longitudinal coordinate $X$ is similar to the time. Thus, stability properties of the symmetric separated flow over slender body are determined by inviscid equations. In this approximation, separation angles ${\theta }_1={\theta }_s\left(X\right)$ and ${\theta }_2=\pi -{\theta }_s$ (Figure 1b) are problem parameters, which may be defined from boundary layer calculations or using experimental data.

For the stability analysis of the flow, we use equations for characteristics of the system (3), which are flow streamlines uniquely corresponding to each solution of Eq. (3). Streamlines are described by the following equations [25]:

$$\frac{\mathit{dZ}}{\mathit{dX}}=w\left(X,Y,Z\right),\frac{\mathit{dY}}{\mathit{dX}}=v\left(X,Y,Z\right),$$

$$X=X_0:Y=Y_0,Z=Z_0.$$E4

The second line of (4) specifies the initial conditions for each streamline at the plane $X=X_0$.

A stability criterion for a solution of the ordinary differential equation system (4) can be studied on the basis of the qualitative analysis of stationary points, that is, using the elementary catastrophe theory [20].

3. Stability analysis of separated conical flows

Exact mathematical results exist only for an autonomous dynamic system which for the considered case corresponds to conical flows. In this case, $B=X$, the cone surface is described by equation $\eta =F\left(z\right)$, $2\delta$ is the maximum cone apex angle, and flow properties depend on conical variables:

$$x=lnX,\eta =\frac{Y}{X},z=\frac{Z}{X}.$$E5

The solution of Euler equation (3) can be found in the class of discontinuous potential functions, so that the following relations are valid:

$$w\left(\eta ,z\right)=\frac{\mathrm{\partial \Phi }}{\partial z},v\left(\eta ,z\right)=\frac{\mathrm{\partial \Phi }}{\partial \eta },$$

$$\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}+\frac{{\partial }^2\mathrm{\Phi }}{\partial {\eta }^2}=0,{\left[\frac{\mathrm{\partial \Phi }}{\partial \eta }-\frac{\mathrm{\partial \Phi }}{\partial z}\frac{\mathit{dF}}{\mathit{dz}}\right]}_{\eta =F\left(z\right)}=F.$$

Here, $\mathrm{\Phi }\left(\eta ,z\right)$ is a flow potential. The equation (5) in conical variables (6) have the following form [20]:

$$\frac{\mathit{dy}}{\mathit{dx}}=v\left(\eta ,z\right)-\eta ,\frac{\mathit{dz}}{\mathit{dx}}=w\left(\eta ,z\right)-z.$$E6

The qualitative behavior of the autonomous dynamic system (7) is determined by properties of singular critical points, where $v\left(X,Y,Z\right)=w\left(X,Y,Z\right)=0$. The bifurcation of a solution arises at appearance of a doubly degenerated critical point, in which the Hessian of system (7) is equal to zero [20], that is,

$$\left(\frac{\partial v}{\partial \eta }-1\right)\left(\frac{\partial w}{\partial z}-1\right)-\frac{\partial v}{\partial z}\frac{\partial w}{\partial \eta }=0.$$E7

Let us consider the flow scheme shown in Figure 1b that corresponds to separated flow at sufficiently large angles of attack. In this vortex flow, the following singular points are present: two vortex centers, the half-saddles separation points at $\theta ={\theta }_s$ and $\pi -{\theta }_s$, attachment points at $\theta =0$ and $\pi$, and the saddle point S with coordinates $\eta ={\eta }_S,z=0$, which is formed at the angle of attack exceeding a critical value. The first six points are Morse saddles in terms of [20], and they do not change their types at the transition to asymmetry; only the saddle point S can change its type. In the vicinity of the point S, taking into account potential and symmetry conditions, the flow velocity components are decomposed to Taylor series:

$$v={\eta }_S-\mathit{d\eta }+b{\eta }^3-3{\mathit{bz}}^2\eta ,w=\mathit{dz}+{\mathit{bz}}^3-3b{\eta }^{2}z,$$

$$y=\eta -{\eta }_S,d\left(\alpha ,{\theta }_s\right)=-\frac{\partial v\left({\eta }_S,0\right)}{\partial \eta }=\frac{\partial w\left({\eta }_S,0\right)}{\partial z},$$

$$b\left(\alpha ,{\theta }_s\right)=\frac{1}{6}\frac{{\partial }^{3}v\left({\eta }_S,0\right)}{\partial {\eta }^3}=\frac{1}{6}\frac{{\partial }^{3}w\left({\eta }_S,0\right)}{\partial z^3},\frac{\partial v\left({\eta }_S,0\right)}{\partial z}=\frac{\partial w\left({\eta }_S,0\right)}{\partial \eta }=0.$$

Near the point S, Eq. (7) are reduced to the form:

$$\frac{\mathit{dy}}{\mathit{dx}}=-\mathit{cy}+{\mathit{by}}^3-3{\mathit{bz}}^{2}y,\frac{\mathit{dz}}{\mathit{dx}}=-c_{1}z+{\mathit{bz}}^3-3{\mathit{by}}^{2}z,$$

$$c_1\left(\alpha ,{\theta }_s\right)=1-d\left(\alpha ,{\theta }_s\right),c\left(\alpha ,{\theta }_s\right)=1+d\left(\alpha ,{\theta }_s\right).$$E8

For Eq. (9), the symmetric flow stability criterion (8) is reduced to the relation:

$$c\left(\alpha ,{\theta }_s\right)c_1\left(\alpha ,{\theta }_s\right)=0.$$E9

Thus, at a bifurcation point, one of the coefficients is equal to zero:$c_1=0$ or c = 0, that corresponds to d = 1 or d = −1.

4. Stability criterion calculation and its verification by comp

To simplify calculations, the separated vortex flow over a slender cone is simulated using the model of point vortexes and conformal mapping of the body cross-section on the unit circle. In this model, the separation vortices and the vortex sheets are mimicked by the point vortices and direct line cuts connecting separation points with these vortices. The Kutta-Zhukovsky condition at the separation points [25] and the equilibrium condition for the vortex system lead to three nonlinear algebraic equations for the vortex intensity $\gamma$ and vortex coordinates ${\eta }_v$, $z_v$:

$$P_i\left(\mathbf{x};{\theta }_s,\alpha \right)=0,i=1,2,3,\mathbf{x}=\left(\gamma ,{\eta }_v,z_v\right).$$

The separation angle ${\theta }_s$ and angle of attack $\alpha$ are parameters in this problem. Calculations were conducted for elliptic cones with the minor axis $\mathit{\lambda X}$, $0<\lambda \le 1$: $\lambda =1$ and 0 correspond to a round cone and a thin delta wing, respectively.

Typical calculations of coefficients $c\left(\alpha \right)$ (Figures 2a and 3a), $c_1\left(\alpha \right)$ and $b\left(\alpha \right)$ (Figures 2b and 3b) for a round cone ($\lambda =1$) are presented in Figure 2 for the laminar boundary layer separation (${\theta }_s\approx 0^{\circ }$) and in Figure 3 for the turbulent boundary layer separation (${\theta }_s\approx {40}^{\circ }$).

Figure 2.

Coefficients of Eq. (8) for round cone ($\lambda =\mathit{1}$) at laminar flow separation (${\theta }_s\approx 0^{\circ }$).

Figure 3.

Coefficients of the Eq. (8) for round cone ($\lambda =\mathit{1}$) at turbulent flow separation (${\theta }_s\approx {\mathit{40}}^{\circ }$).

Parametric studies showed that coefficients $c_1\left(\alpha ,{\theta }_s\right)$ and $b\left(\alpha ,{\theta }_s\right)$ are always positive, and only $c\left(\alpha ,{\theta }_s\right)$ changes its sign for each separation angle when the angle of attack is varied. From these properties, it follows that bifurcation criterion can be written as follows:

$$c\left(\alpha ,{\theta }_s\right)=0.$$E10

In Figure 4a and b, calculated curves $c\left(\alpha ,{\theta }_s\right)=0$ for round ($\lambda =1$, Figure 4a) and elliptic (in Figure 4b, line 1 corresponds to $\lambda =$ 0.3, and line 2 corresponds to $\lambda =$ 0.5) cones are shown by solid lines. Symbols correspond to experimental data of asymmetric state origin; vertical and horizontal segments show measurement accuracy of normalized angle of attack and separation angle.

Figure 4.

The stability criteria $c\left(\alpha ,{\theta }_s\right)=\mathit{0}$ for round (a) and elliptic (b) cones.

Only data for homogeneous flow conditions (laminar or turbulent) are used for the calculation verification, a transition regime is excluded according to the state diagram similar to the diagram [9] for the body of the type revived cylinder. For elliptic cones, experimental data with the flow asymmetry due to the vortex explosion [10] were excluded. Data [4] for the turbulent separation on the round cone with $\delta =$5⁰ agree with data [5] obtained at the large Reynolds number (13.5∙10⁶) and with other data obtained in low noise wind TunnelSat large Reynolds numbers [8, 9, 21, 22, 23]. Experiments [21, 22, 23] were performed in the low noise wind tunnel of ITAM SB RAS at high level of the cone surface smoothness; the turbulent flow was generated by turbulators.

The utilized asymptotic approach is appropriate at the limit $\delta \to 0$. The data for round cones of different apex angles in Figure 4a (○ − $\delta =$10^°, ▲ − $\delta =$5^°, ● − $\delta =$3^°) [4] demonstrate the well convergence of the theory and experiments with the cone apex angle diminishing for the turbulent boundary layer separation (${\theta }_s\approx {40}^{\circ }$). For the thinner cone ($\delta =$3^°), experimental results are absent; in this case, the extrapolation of [4] is used, and the obtained value is agreed well with calculation results. Calculations conform well by experiments [21, 22, 23] (Figure 4a, ◊ – $\delta =$5^°), as for turbulent (${\theta }_s\approx {40}^{\circ }$) and the laminar (${\theta }_s\approx 0^{\circ }$) separations. In spite of the rough enough flow model and large apex angle ($\delta ={10}^{\circ }$), the calculated stability criteria in Figure 4b for elliptic cones with $\lambda =$ 0.3 (1) and $\lambda =$ 0.5 (2) are consistent satisfactorily with experiments [4].

It should be noted that, in contrast to presented results, the bifurcation diagram of the intrinsic asymmetric solutions of the model [17] does not correspond the experimental data, the saddle point appears in the flow at significantly large angles [1, 2, 3, 4, 21, 22, 23].

5. Transition to asymmetry local analysis

Although the catastrophe theory [20] gives only the criterion of the asymmetry origin but cannot describe the transition process, some qualitative results of the beginning of this process can be obtained by the analysis of local streamline Eq. (8). The transition to asymmetric state is associated with the typical change of the critical point S, when the coefficient $c\left(\alpha ,{\theta }_s\right)$ changes its sign, and with the global flow instability at $c\left(\alpha ,{\theta }_s\right)$ < 0. Calculations for elliptic cones show that b > 0, $c_1>3c$. Thus, critical points of Eq. (8) are as follows: the symmetric saddle point $y=z=0$ (S), and also points $S_1$, $S_2$ at $c<0$ and $S_3$, $S_4$ at $c>0$:

$$c<0.S_1:y=0,z=\sqrt{\frac{c_1}{b}};S_2:y=0,z=-\sqrt{\frac{c_1}{b}},$$

$$c>0.S_3:z=0,y=\sqrt{\frac{c}{b}};S_4:z=0,y=-\sqrt{\frac{c}{b}}.$$

At $c<0$, three nondegenerate critical points ($S$, $S_1$, and$S_2$) exist at the axis $y=0$; S is unstable saddle. At $c=0$_, critical points S, S₃, and S₄ coincide, that is, S is triply degenerated in the $y$-direction and has the saddle-node type. At $c>0$, there are five nondegenerate critical points: $S$; $S_1$ and $S_2$ at the axis $y=0$, and S₃ and S₄ at the axis $z=0$; in this case, $S$ is a stable node.

For qualitative analysis of transition to asymmetry, let us consider the first equation of (6) in the plane $z=0$, where it is reduced to the simplest form of the Ginzburg-Landau equation [25]:

$$\frac{\mathit{dy}}{\mathit{dx}}=-\mathit{cy}+{\mathit{by}}^3.$$E11

At $c\ne 0$ and $b>0$_,Eq. (11) has the following solution:

$$h\left(X\right)=y^2=h_c{\left[1-B_0{\left(\frac{X}{X_0}\right)}^{2c_1}\right]}^{-1},B_0=\frac{h_0-h_c}{h_0},h_c=\frac{c}{b}.$$E12

The condition $h\left(X_0\right)=h_0$ specifies an initial streamline perturbation in the plane $z=0$.

At $c>0$, the solution of Eq. (12) is stable in the linear approximation ($b=0$):

$$y=y_0{\left(\frac{X}{X_0}\right)}^{-c}\to 0,X\to \infty .$$

For $h_0<h_c$_, the coefficient $B_0<0$ and the solution (12) of the nonlinear Eq. (11) is also stable:$h\left(X\right)\to 0$ at $X\to \infty$, that is, streamlines remain in the attraction region of the stable node $S$.

For $h_0>h_c$_, the coefficient $B_0>0$ and $h\left(X\right)\to \infty$ at $X\to X_c$, where

$$X_c=X_0{B}_0^{-\frac{1}{2c}},Y_c=\pm X_0\sqrt{\frac{c}{b}}=\pm X_0\sqrt{h_c}.$$

Here, $X_c$ is length of perturbation development; it is a linear function of the distance from the initial data plane to the top of the body $X_0$ and nonlinear function of the initial perturbation amplitude $h_0$. Thus, at $h_0>h_c$ and $c>0$, streamlines leave the attraction region of the stable node $S$ at the finite distance $X=X_c$ from the cone apex. The value $Y_c$ characterizes the threshold level of stationary perturbation at the plane $X=X_0$ that initiates instability. $Y_c$ and $X_c$ are decreased when the initial data plane is shifted to the cone apex: $Y_c\to 0$ and $X_c\to 0$ at $X_0\to 0$. Therefore, at $h_0>h_c$ and $c>0$_, the solution (12) describes the subcritical bifurcation under the influence of finite perturbations.

If $c<0$_, the solution (12) is unstable in linear and nonlinear approximations since the coefficient $B_0>0$ and $h\left(X\right)\to \infty$ at $X\to X_c$.

For $c=0$_, the solution of Eq. (12) has the following form:

$$h=h_0{\left[1-2{\mathit{bh}}_{0}ln\left(\frac{X}{X_0}\right)\right]}^{-1},X_c=X_{0}exp\left(\frac{1}{2{\mathit{bh}}_0}\right)$$

This is also nonlinear unstable solution, in which the transition region depends on the initial perturbation exponentially.

The presented results show that the asymmetry origin mechanism is essentially nonlinear. At angles of attack less the critical value, the subcritical bifurcation is possible if perturbations are larger than threshold value, which is lower than closer to the top it arises. Due to the nonlinearity, it is impossible to extract perturbations from general equations and study them separately. There is a finite length where perturbations are amplified to infinity, and this distance is diminished if the initial data plane is shifted to the body apex. The subcritical bifurcation is possible at angles of attack lower than a critical value, if the initial perturbation exceeds a threshold, which decreases with shifting of the initial data plane to the body apex.

The mechanism of asymmetric flow arising has properties both convective and absolute instability, where the coordinate $X$ is treated as a time. However, this terminology relates to the linear theory of hydrodynamic stability and cannot fully describe the considered nonlinear process. This instability can be characterized more exactly as global or structural instability. The presented qualitative analysis and the asymmetry origin criterion allow to explain most of the effects found in experiments and numerical simulations. The developed approach allows also to analyze unsteady perturbations.

6. Asymmetry control using plasma discharge and local surface heating

For analysis of these problems, calculations of 3D turbulent boundary layer equations in the local self-similar approximation on a round slender cone with the half angle at the top δ at the angle of attack ${\alpha }^{\ast }=\mathit{\alpha \delta }$ are utilized. The plasma discharge effect is modeled by the volume heat source in the boundary layer. The effect of the gas ionization is neglected since the ionization coefficient is small, of the order of 10⁻⁵. The flow scheme and the coordinate system are shown in Figure 5. As it is seen in Figure 5b, the heat release acts on the boundary layer as an obstacle.

Figure 5.

Plasma discharge in the boundary layer on a round cone.

Boundary layer equations are simplified using the Dorodnitsyn-Stewardson transformation, longitudinal $f\left(x,\eta ,\phi \right)$ and transverse $g\left(x,\eta ,\phi \right)$ stream functions. Transformations have a form:

$$\mathit{d\eta }=\frac{\rho }{\sqrt{x}}\mathit{dy},y=\sqrt{Re}\frac{y^{\ast }}{l},Re=\frac{{\rho }_{\infty }{u}_{\infty }l}{{\mu }_{\infty }},\rho =\frac{{\rho }^{\ast }}{{\rho }_{\infty }},$$

$$V=-\left(\frac{3}{2}f+\frac{\partial w_e}{\partial \varphi }g+w_e\frac{\partial g}{\partial \varphi }\right),U=\frac{\partial f}{\partial \eta }=\frac{u^{\ast }}{U_{\infty }},W=\frac{\partial g}{\partial \eta }=\frac{w^{\ast }}{\delta U_{\infty }{w}_e}h=\frac{h^{\ast }}{h_{\infty }{h}_e}.$$

Here, $U\left(x,\eta ,\phi \right)$ and $W\left(x,\eta ,\phi \right)$ are nondimensional longitudinal (along the cone surface) and circumferential velocities, $\rho \left(x,\eta ,\phi \right)$ and $h\left(x,\eta ,\phi \right)$ are nondimensional density and enthalpy related to their values on the outer boundary, $V\left(x,\eta ,\phi \right)$ is transformed normal to the cone surface velocity, $\phi =\theta +0.5\pi$ is angle of cylindrical coordinates, indexes ∞ and e correspond to parameters in the freestream flow and on the outer boundary. The circular inviscid velocity is approximated by the formula $w_e=2\alpha sin\phi$.

Self-similar boundary-layer equations and boundary conditions are as follows:

$$V\frac{\partial U}{\partial \eta }+w_{e}W\frac{\partial U}{\partial \phi }=\frac{\partial }{\partial \eta }\left(m\frac{\partial U}{\partial \eta }\right),$$

$$V\frac{\partial W}{\partial \eta }+w_{e}W\frac{\partial W}{\partial \phi }+\mathit{UW}+\frac{\partial w_e}{\partial \phi }{W}^2-h\left(1+\frac{\partial w_e}{\partial \phi }\right)=\frac{\partial }{\partial \eta }\left(m\frac{\partial W}{\partial \eta }\right),$$

$$V\frac{\partial h}{\partial \eta }+w_{e}W\frac{\partial h}{\partial \phi }-Q\left(x,y,\phi \right)=\frac{\partial }{\partial \eta }k\frac{\partial h}{\partial \eta },\mathit{\rho h}=1,$$

$$\eta =\infty :U=W=h=1,\eta =0:U=W=0,h=h_w\left(\frac{\partial h}{\partial \eta }=0\right).$$E13

For turbulent flows, the transformed heat conductivity $k$ and the transformed viscosity $m$ are determined using the Cebeci-Smith model:

$$m=\mathit{\rho \mu }\left(1+\epsilon \right),k=\frac{\mathit{\rho \mu }}{Pr}\left(1+\frac{Pr}{{Pr}_t}\epsilon \right);0\le y\le y_w:\epsilon ={\epsilon }_1{\left(\frac{\partial U}{\partial \eta }\right)}^2,$$

$$y>y_w:\epsilon =0.0168\sqrt{Rex}{D}_1\frac{{\delta }_u}{\mathit{h\mu }};{\epsilon }_1=0.16D_1\frac{\sqrt{Rex}}{\mu }{\left(\frac{\mathit{eD}}{h}\right)}^2,e=\underset{0}{\overset{y_e}{\int }}\mathit{hdy},$$

$${\delta }_u=\underset{0}{\overset{y_e}{\int }}\left(1-U\right)\mathit{dy},D=1-exp\left(-d\right),d={\left(Rex\right)}^{\frac{1}{4}}\frac{e}{26h_w}\sqrt{\frac{1}{{\mu }_w}\frac{\partial U}{\partial y}};$$

$$D_1=\left[1+{\left(\frac{e}{e_e}\right)}^6\right],\mu =h^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{1+h_S}{h+h_S},h_S=\frac{110.4}{T_{\infty }}.$$

Here, ${Pr}_t\approx 0,9$

is the turbulent Prandtl number, and the subscript $w$ corresponds to parameters on the wall. The heat source term Q in the left-hand side of the energy Eq. (13) describing the discharge heat release is modeled as follows:

$$Q=\frac{Q^{\ast }\mathit{xl}}{h_{\infty }{U}_{\infty }}=Q_0{y}^{2}exp\left[-\frac{{\left(y-y_c\left(\phi \right)\right)}^2}{\sigma }\right],y_c=2y_0\sqrt{\left|\left(\phi -{\phi }_1\right)\left({\phi }_2-\phi \right)\right|}.$$

Here, $Q^{\ast }$ and $Q_0$ are a dimensional source intensity and its maximum; $\sigma$ characterizes the discharge width, $y_c\left(\phi \right)$ is a centerline of the discharge that is approximated by the parabola; $y_0$ is a maximum distance from the discharge centerline to the wall; and the angles ${\phi }_1$ and ${\phi }_2$ determine electrode locations. The total power released in the boundary layer is determined as follows:

$$P=\underset{V_h}{\iiint }{\rho }^{\ast }{Q}^{\ast }{a}^{\ast }\left(x\right){\mathit{dx}}^{\ast }\mathit{d\varphi }{\mathit{dy}}^{\ast }=\frac{{\mathit{\beta \rho }}_{\infty }{h}_{\infty }{U}_{\infty }{l}^2}{\sqrt{Re}}\underset{x_1}{\overset{x_2}{\int }}\sqrt{x}\mathit{dx}\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}\mathit{d\phi }\underset{0}{\overset{\infty }{\int }}\mathit{Qdy}.$$E14

If $Q_0$ is a function of the coordinate x and the flow is turbulent, then the boundary layer is not self-similar. However, we can use Eq. (13) for the first-order estimations of flow characteristics. Some calculation results are presented in Figure 6 for the following parameters: $\alpha =3.15$, $\delta =5^{\circ },l=1$ m, $T_{\infty }={288.15}^{\circ }K,U_{\infty }=10$ m/s, $\sigma =1$, $y_0=1$; the center between electrodes is located at ${\varphi }_0=.5\left({\varphi }_1+{\varphi }_2\right)=1.714$ rad (98.25°); ${\varphi }_1={\varphi }_0-3\mathrm{\Delta }\varphi$, ${\varphi }_2={\varphi }_0+3\mathrm{\Delta }\varphi$, where $\mathrm{\Delta }\varphi =0.0314159$ is the integration step of the finite-difference approximation.

Figure 6.

Profiles of the temperature and the circumferential velocity across the boundary layer in the discharge region.

Figure 7a demonstrates the plasma discharge effect on the separation point. As the heat-source intensity increases from 0 to 400, the separation angle, ${\phi }_s$, decreases from ${133}^{\circ }$ to about ${105}^{\circ }$. It is seen that the plasma heating is more effective in the range $Q_0<100$, where the slope $d{\phi }_s/{\mathit{dQ}}_0$ is relatively large.

Figure 7.

Discharge effect on the separation angle and flow state.

Figure 7b illustrates feasibility of the vortex structure control using a local volume boundary-layer heating. The solid line in Figure 4 represents boundary between symmetric and asymmetric flow states. Due to the heat source, the flow configuration changes from the initial asymmetric state (${\phi }_s\approx {133}^{\circ }$, symbol 1) to the symmetric state with ${\phi }_s\approx {120}^{\circ }$ (symbol 2). This requires a nondimensional heat source intensity $Q_0\approx 30$. Using Eq. (14), we estimate that the corresponding total power is approximately equal to 480 W. This example indicates that the method is feasible for practical applications of the global flow structure control [21, 22, 23].

Another method to control the flow separation based on the body surface local heating is also analyzed for the case of a slender round cone. The boundary layer characteristics were calculated using Eq. (13) with $Q=0$ and with the boundary condition for enthalpy $h\left(0,\phi \right)=h_w$, where $h_w$ is a normalized wall temperature. The flow scheme for this case is shown in Figure 8. The whole cone surface had the temperature $T_w=T_{\infty }$ ($h_w=1$) except the narrow strip $\mathrm{\Delta }\phi$ with the temperature $T_w=T_{\mathit{\text{strip}}}>T_{\infty }$. The boundary layer calculations were conducted for the cone of length l = 1 m with apex angle $2\delta =$10° at the freestream velocity 20 m/s and normalized angles of attack $\alpha =2-4.$ The freestream temperature was $T_{\infty }$= 0°C, the strip temperature varied in the range of $T_{п}=100-800$°C, the strip width was $\mathrm{\Delta }\phi =$12°, and its center was located at $\phi =$110°.

Figure 8.

Effect of heated surface strip on the separation.

The separation angles ${\phi }_s$ as functions of strip temperature $T_{\mathit{\text{strip}}}$ are presented in Figure 8b for $\alpha =$3 and $\alpha =$4. For $\alpha =$2, the separation angle does not depend on strip temperature. At $\alpha =$3, the separation angle ${\phi }_s$ decreases from 133° to 122° with strip temperature increasing to 200°С. The separation angle is constant up to $T_{\mathit{\text{strip}}}=$ 600°С, and then it varies to ${\phi }_{1s}\approx$ 109° as strip temperature increases to 700°С and stabilizes at this value at the subsequent strip temperature growth. For $\alpha =$4, the separation angle decreases monotonically to ${\phi }_{1s}\approx$ 109°, if the strip is heated from 100°С to 600°С and remains constant at further temperature growth.

The results presented in this section show that the local surface heating can be used for control of separation location; however, this method is less effective than the plasma discharge.

7. Conclusions

In this paper, two main problems related with the asymmetry arising at separated vortex flow over thin bodies are considered. In the first one, the new approach for the transition analysis to the asymmetric state appearance is proposed, and its verification on the base of the comparison of calculations with experimental results is done. In the other part, on the basis of the turbulent boundary layer, calculations and the new model for the plasma discharge possibilities of the asymmetry control using volume gas or local surface heating were considered.

The criterion of the transition to the asymmetry was obtained by applying the catastrophe theory to the analysis of the saddle critical point of the dynamical system describing symmetric separated flow streamlines. The qualitative analysis of the transition to asymmetric state properties based on the symmetric flow streamline behavior under perturbations studies near the saddle point is presented. The analysis shows that the transition is a nonlinear process, and it is characterized by the finite transition length and the threshold level of perturbations, which are diminished if the initial data plane is shifted to the cone apex. Theoretical results are verified by comparison with experimental data for laminar and turbulent flows over round and elliptic pointed cones. As shown, these results help to explain most of experimental observations and numerical simulations.

A new theoretical plasma discharge model as a local gas heat source was presented. The performed studies of separation angle control in a turbulent boundary layer using the local volume and surface gas heating show that both methods can be used for flow global structure control, but the first one is more effective.