4. Studies of the symmetric flow instability over thin bodies and the control possibility on the base of the interaction model of 3D boundary layer with the electrical discharge

The electric discharge is considered as one of effective methods for control of the flow asymmetry over bodies [23–27]. However, to select optimal control parameters, it needs to have a reasonable criterion for the asymmetry origin and a possibility for fast estimation of the control effect. For the second problem, the model of the boundary layer and discharge interaction is proposed. The scheme of this model is shown in Figure 8 [28–31, 37].

heat intensity distribution across the boundary layer. The temperature reaches the maximum value near the rear electrode, φ ¼ φ<sup>2</sup> ¼ 1:809. Behind the heat source region, the temperature maximum decreases and moves toward the upper boundary-layer edge due to the heat diffusion. The station φ ¼ 1:87 is located just after the

Profiles of temperature (а) и and circumferential velocity (b) across the boundary layer near the heat-release

Figure 10a demonstrates the plasma discharge effect on the separation point. As the heat source intensity increases from 0 to 400, the separation angle, φs, decreases from 133° to about 105°. It is seen that the plasma heating is more effective in the

Figure 10b illustrates feasibility of the vortex structure control using a local boundary-layer heating on the base of the developed criterion of symmetric flow stability (solid line). Due to the heat release, the flow configuration changes from the initial asymmetric state (φ<sup>s</sup> ≈133°, symbol 1) to the symmetric state with θ<sup>s</sup> ≈ 120° (symbol 2). This requires a nondimensional heat source intensity Q<sup>0</sup> ≈ 30 that corresponds to the total power which is approximately equal to 480 W. This example indicates that the method is feasible for practical applications of the global

The method of the global flow stability was developed [27–31] using the asymptotic approach for the flow over slender cones, the separated inviscid flow model [34] and the stability theory of autonomous dynamical systems [35]. Comparison of the calculated criteria for different elliptic slender cones with experimental data for

range Q<sup>0</sup> , 100, where the slope dφs=dQ<sup>0</sup> is relatively large.

laminar and turbulent boundary layers sowed its efficiency.

Discharge effect on the separation angle (a) and flow state (b).

separation point

3D Boundary Layer Theory

DOI: http://dx.doi.org/10.5772/intechopen.83519

Figure 9.

region.

flow structure control.

Figure 10.

19

It is assumed the plasma discharge effect can be modeled by the heat source in the boundary layer. The effect of gas ionization is neglected since the ionization coefficient is of the order of 10�<sup>5</sup> . This source in the energy equation is presented by formulas:

$$Q = \frac{Q^\* x l}{h\_{\infty} u\_{\infty}} = Q\_0 y^2 \exp\left[-\frac{(y - \mathcal{y}\_c(\rho))^2}{\sigma}\right], \quad \mathcal{y}\_c = 2\mathcal{y}\_0 \sqrt{| (\rho - \mathcal{q}\_1)(\rho\_2 - \rho) |} \tag{38}$$

Here Q<sup>∗</sup> is a dimensional source intensity; Q<sup>0</sup> is a maximum of dimensionless heat-release intensity; σ characterizes the discharge width; ycð Þ φ is a centerline of the discharge that is approximated by the parabola; y<sup>0</sup> is a maximum distance from the discharge centerline to the wall; and the angles φ<sup>1</sup> and φ<sup>2</sup> determine the electrode locations.

Calculations of the turbulent boundary layer characteristics were conducted using the method [10] for a slender cone of half-apex angle <sup>δ</sup><sup>c</sup> <sup>¼</sup> <sup>5</sup><sup>∘</sup> at the angle of attack <sup>α</sup> <sup>¼</sup> <sup>α</sup><sup>∗</sup>=δ<sup>c</sup> <sup>¼</sup> <sup>3</sup>:15. Other parameters are: <sup>l</sup> <sup>¼</sup> 1 m, <sup>T</sup><sup>∞</sup> <sup>¼</sup> 288K, <sup>u</sup><sup>∞</sup> <sup>¼</sup> 10 m=s, σ ¼ 1, and y<sup>0</sup> ¼ 1; the center between electrodes is located at ϕ<sup>0</sup> ¼ 0:5 ϕ<sup>1</sup> þ ϕ<sup>2</sup> ð Þ¼ <sup>1</sup>:714 rad 98:25° � �, <sup>ϕ</sup><sup>1</sup> <sup>¼</sup> <sup>ϕ</sup><sup>0</sup> � <sup>3</sup>Δϕ, and <sup>ϕ</sup><sup>2</sup> <sup>¼</sup> <sup>ϕ</sup><sup>0</sup> <sup>þ</sup> <sup>3</sup>Δϕ, where <sup>Δ</sup><sup>ϕ</sup> <sup>¼</sup> <sup>0</sup>:0314159 is the integration step of the finite-difference approximation.

In Figure 9, the dimensionless enthalpy (Figure 9a) and circumferential velocity (Figure 9b) profiles across the boundary layer are shown as functions of η for Q<sup>0</sup> ¼ 200 and for different polar angles φ. These profiles are similar to the source

Figure 8. A scheme of discharge interaction with the boundary layer.

Figure 9.

m = 3/4 and 7/8 in relation to cases а and b, respectively. The longitudinal velocity

Near-wall singularities generate the flow structure including three asymptotic sublayers describing the viscous-inviscid interaction similar as near the 2D separation point. However, the viscous-inviscid interaction is not enough to remove the singularity of the obtained type. Near the wall sublayer close to the symmetry plane the fourth region is formed, in which the flow is described by the parabolized Navier-Stocks equations similar to the above case of the outer singularity.

4. Studies of the symmetric flow instability over thin bodies and the control possibility on the base of the interaction model of 3D boundary

The electric discharge is considered as one of effective methods for control of the flow asymmetry over bodies [23–27]. However, to select optimal control parameters, it needs to have a reasonable criterion for the asymmetry origin and a possibility for fast estimation of the control effect. For the second problem, the model of the boundary layer and discharge interaction is proposed. The scheme of this model

It is assumed the plasma discharge effect can be modeled by the heat source in the boundary layer. The effect of gas ionization is neglected since the ionization

Here Q<sup>∗</sup> is a dimensional source intensity; Q<sup>0</sup> is a maximum of dimensionless heat-release intensity; σ characterizes the discharge width; ycð Þ φ is a centerline of the discharge that is approximated by the parabola; y<sup>0</sup> is a maximum distance from the discharge centerline to the wall; and the angles φ<sup>1</sup> and φ<sup>2</sup> determine the elec-

Calculations of the turbulent boundary layer characteristics were conducted using the method [10] for a slender cone of half-apex angle <sup>δ</sup><sup>c</sup> <sup>¼</sup> <sup>5</sup><sup>∘</sup> at the angle of attack <sup>α</sup> <sup>¼</sup> <sup>α</sup><sup>∗</sup>=δ<sup>c</sup> <sup>¼</sup> <sup>3</sup>:15. Other parameters are: <sup>l</sup> <sup>¼</sup> 1 m, <sup>T</sup><sup>∞</sup> <sup>¼</sup> 288K, <sup>u</sup><sup>∞</sup> <sup>¼</sup> 10 m=s, σ ¼ 1, and y<sup>0</sup> ¼ 1; the center between electrodes is located at ϕ<sup>0</sup> ¼ 0:5 ϕ<sup>1</sup> þ ϕ<sup>2</sup> ð Þ¼ <sup>1</sup>:714 rad 98:25° � �, <sup>ϕ</sup><sup>1</sup> <sup>¼</sup> <sup>ϕ</sup><sup>0</sup> � <sup>3</sup>Δϕ, and <sup>ϕ</sup><sup>2</sup> <sup>¼</sup> <sup>ϕ</sup><sup>0</sup> <sup>þ</sup> <sup>3</sup>Δϕ, where <sup>Δ</sup><sup>ϕ</sup> <sup>¼</sup> <sup>0</sup>:0314159 is

In Figure 9, the dimensionless enthalpy (Figure 9a) and circumferential velocity (Figure 9b) profiles across the boundary layer are shown as functions of η for Q<sup>0</sup> ¼ 200 and for different polar angles φ. These profiles are similar to the source

, yc ¼ 2y<sup>0</sup>

. This source in the energy equation is presented by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>φ</sup> � <sup>φ</sup><sup>1</sup> j j ð Þð Þ <sup>φ</sup><sup>2</sup> � <sup>φ</sup> <sup>p</sup> (38)

layer with the electrical discharge

is shown in Figure 8 [28–31, 37].

coefficient is of the order of 10�<sup>5</sup>

<sup>¼</sup> <sup>Q</sup>0y<sup>2</sup> exp � <sup>y</sup> � ycð Þ <sup>φ</sup> � �<sup>2</sup>

the integration step of the finite-difference approximation.

A scheme of discharge interaction with the boundary layer.

σ " #

formulas:

<sup>Q</sup> <sup>¼</sup> <sup>Q</sup>∗xl h∞u<sup>∞</sup>

trode locations.

Figure 8.

18

perturbation singularity is related only with the near-wall singularity.

Boundary Layer Flows - Theory, Applications and Numerical Methods

Profiles of temperature (а) и and circumferential velocity (b) across the boundary layer near the heat-release region.

heat intensity distribution across the boundary layer. The temperature reaches the maximum value near the rear electrode, φ ¼ φ<sup>2</sup> ¼ 1:809. Behind the heat source region, the temperature maximum decreases and moves toward the upper boundary-layer edge due to the heat diffusion. The station φ ¼ 1:87 is located just after the separation point

Figure 10a demonstrates the plasma discharge effect on the separation point. As the heat source intensity increases from 0 to 400, the separation angle, φs, decreases from 133° to about 105°. It is seen that the plasma heating is more effective in the range Q<sup>0</sup> , 100, where the slope dφs=dQ<sup>0</sup> is relatively large.

Figure 10b illustrates feasibility of the vortex structure control using a local boundary-layer heating on the base of the developed criterion of symmetric flow stability (solid line). Due to the heat release, the flow configuration changes from the initial asymmetric state (φ<sup>s</sup> ≈133°, symbol 1) to the symmetric state with θ<sup>s</sup> ≈ 120° (symbol 2). This requires a nondimensional heat source intensity Q<sup>0</sup> ≈ 30 that corresponds to the total power which is approximately equal to 480 W. This example indicates that the method is feasible for practical applications of the global flow structure control.

The method of the global flow stability was developed [27–31] using the asymptotic approach for the flow over slender cones, the separated inviscid flow model [34] and the stability theory of autonomous dynamical systems [35]. Comparison of the calculated criteria for different elliptic slender cones with experimental data for laminar and turbulent boundary layers sowed its efficiency.

Figure 10. Discharge effect on the separation angle (a) and flow state (b).
