ISSN 0015-4628, Fluid Dynamics, 2014, Vol. 49, No. 4, pp. 547–556. © Pleiades Publishing, Ltd., 2014.
Original Russian Text © A.B. Vatazhin, A.Yu. Makarov, V.A. Stepanov, 2014, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2014, Vol. 49, No. 4, pp. 143–154.
Approximate Calculation of Channel Flows under Force and Energy Actions
A. B. Vatazhin, A. Yu. Makarov, and V. A. Stepanov
Baranov Central Institute of Aviation Motors (CIAM), ul. Aviamotornaya 2, Moscow, 111116 Russia e-mail: email@example.com
Received September 25, 2013
Abstract—Gasdynamic channel flows under force and energy actions are considered. An approximate method is proposed for solving the gasdynamic equations that describe these flows. The method includes the separation of an “active” flow volume in which the integral electric force and applied power, whose densities are assumed to be uniform, are concentrated and the numerical integration of the system of hydrodynamic equations over the entire channel (in laminar and turbulent variants) with the piecewise constant force and energy sources obtained. The results of experimental investigation are presented for the flow that arises after two accessories mounted on the opposite walls of the vertical rectangular channel of constant cross-section, which create a dielectric barrier discharge (DBD actuators). This flow is numerically simulated using the method developed. On the basis of the method proposed the flow characteristics are determined for a model subsonic diffuser on whose lower wall, immediately in front of the separation zone, the DBD actuator is mounted. The efficiency of this accessory in reducing the gasdynamic losses is demonstrated.
Keywords: gasdynamic channel flows, force and energy actions, dielectric barrier discharge (DBD),
DBD effect on channel flows.
Currently, the possibility to use electric final-control elements, for example, the dielectric barrier discharge, for the action on gasdynamic flows is intensely studied. The interest in using the accessories based on dielectric barrier discharge for gasdynamic applications is caused by their certain advantages: low power consumption, relatively small weight, the possibility to mount the discharge electrodes almost aflush with the walls without introducing additional perturbations into the flow, and, most important, the possibility to mount the DBD actuator on that place of the surface where adverse effects (origination of perturbations, flow separation, etc.) begin to develop.
The experimental and theoretical respects of interaction of single DBD actuators and their systems on various gasdynamic flows were considered in numerous studies. (For example, mention recent publications [1–6].) However, the actions on the flow presented in these studies mainly relate to outer gasdynamic flows.
Analysis of inner flows in the presence of DBD is, in many respects, a new problem which is important for aircraft and engine applications. The present study is related to this problem.
In Sect. 1 the general system of gasdynamic equations is presented with account for force and energy actions exerted by electric discharges. Obtaining specific expressions for force and heat sources requires complicated theoretical and calculational investigations and the construction of adequate physical and mathematical models, which is now being realized in Russia and abroad [1–6]. However, this work is far from completion.
Therefore, in Sect. 2 we propose an approximate method of estimating the efficiency of DBD effect on the gasdynamic flow by separating an “active” volume in which the integral DBD-generated electric force and applied power, whose densities are assumed to be uniform, are concentrated. The system of 547 548 VATAZHIN et al. gasdynamic equations is then integrated numerically (in laminar and turbulent variants) with the obtained piecewise-constant force and energy sources. The correct choice of the “active” volume and the integral force and power is important and can be done using experiments and qualitative consideration.
In Sect. 3, the results of experimental investigation of the flow that arises in the vertical channel of constant cross-section in the presence of two DBD actuators are presented. The most important feature of this study consisted in measuring by a weight sensor the channel draught that arises. This enabled us to approximately determine in Sect 4 the integral electric force that acts on the flow and to calculate the flow using the above-described method.
In order to understand the future trends of using the DBD-systems for controlling channel flows in the presence of separated zones (in diffusers) we present in Sect. 5 the results of numerical simulation of such flows on the basis of the approximate method proposed. 1. GASDYNAMIC EQUATIONS WITH FORCE
AND ENERGY SOURCES OF ELECTRIC NATURE
In the general case, the continuity, momentum and energy equations have the form dρ dt + ρ divv = 0, (1.1) ρ dvdt =−∇p + ∂τik ∂xk ei + f, (1.2) ρ ddt ( e + v2 2 ) =−div(pv) + ∂∂xk ( τikvi ) − divq + N, τik = μ ( ∂vi ∂xk + ∂vk ∂xi − 2 3 δik ∂vl ∂xl ) + ζδik ∂vl∂xl .
Here, p, ρ , and e are pressure, density, and the internal energy per unit medium mass; v and vi (i, k = 1, 2, 3) the medium velocity vector and its components; τik the viscous stress tensor and its components; μ and ς the first and second viscosity coefficients; q is the heat flux density vector; f the electric force per unit medium volume; N the electric energy released per unit time in unit volume; t time; ei are unit basis vectors of the Cartesian coordinate system xi; and δik is the Kronecker tensor. For recurring indices the summation convention applies.
From the above system there follows the energy equation for enthalpy h ρ dhdt = d p dt + τik ∂vi ∂xk − divq + D, h = e + pρ , D = N − fv. (1.3)
The differential equations presented can be rewritten in the integral form, which is, strictly speaking, initial. For the channel with impermeable wall and inlet and outlet cross-sections 1 and 2 the integral momentum equation and the integral equation for stagnation enthalpy h∗ have the form ∫ 2 ρvvn dΣ − ∫ 1 ρvvn dΣ =− ∫