The application of the projection method for constructing the shape of a body with minimum drag

The problem is considered of constructing a semi-infinite axisymmetric body with minimum drag in subsonic flow of an ideal gas. This problem is formulated as the problem of finite-dimensional minimization by prescribing the shape of the body in parametric form and applying the projection method for solving a flow problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 108–113, March–April, 1985.     

Notes on gas–dynamic design of supersonic flying vehicles

It is shown that the lift–to–drag ratio of a thin delta wing is significantly lower than the lift–to–drag ratio of an infinitely long swept plate with an identical lift force. The effect of sweep on a finite wing may be used by excluding disturbances from the leading edge of the wing via introducing a hardened stream surface (wedge) and increasing the wing length. A three–shock waverider is proposed for choosing the optimal parameters. The sharp wedge may be avoided by replacing planar shock waves by a cylindrical shock wave upstream of the blunted wedge. If the leading edge of the wedge is not parallel to the rib that is a source of the expansion wave, a plate with zero wave drag, generating a lift force, may be obtained behind this rib. The system of regularly intersecting shock waves may be applied to design a forward–swept wing.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Wings with optimal characteristics in hypersonic flow

We present some exact results and results obtained by the local variation method [1–3] from numerical solutions of variational problems on the symmetric wing of minimum drag and the wing of maximum L/D in hypersonic flow; a modification of the method of local variations is proposed for the numerical solution of variational problems with isoperimetric constraints. The Newton method [4] was used to calculate the pressure distribution over the wing surface.The author wishes to thank M. N. Kogan and O. G. Fridlender for many helpful discussions.     

Optimization of laminar-turbulent transition delay by means of local heating of the surface

The stability and position of laminar-turbulent transition in the boundary layer on a body heated near the leading edge are analyzed. The point of transition is found using the linear theory of the stability of plane-parallel flow and thee ^N -method. It is shown that by heating a tiny area near the leading edge to a temperature exceeding that of the oncoming flow by a factor of two to four, transition may be delayed, even on a thermally insulated surface. For highly radiating surfaces the energy saved by reducing the friction drag may exceed the heating energy by a factor of three. It is shown that by varying the pressure distribution and surface heating it is possible either to increase the airfoil lift for a fixed transition point or delay transition for a fixed lift.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–99, July–August, 1995.     

Least-drag bodies moving in media subject to the locality hypothesis

A solution is constructed for the variational problem of a body of minimum drag moving at constant velocity in media in which under certain assumptions the force exerted by the medium on an area element of the body surface depends only on its orientation relative to the direction of motion (locality hypothesis). The representations of the normal (pressure) and tangential (friction) components of the force embrace a broad range of the conditions realized in the motion of a body through gases and dense media.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 95–106, January–February, 1992.     

Hypersonic flow past a rectangular profile through which gas is blown strongly

In the present paper, we consider the hypersonic flow past a rectangular profile and the end of a cylinder when there is strong distributed blowing of gas through their flat front parts. The injected gas is assumed to be inviscid, and the pressure on the contact surface which separates the exterior flow and the blowing layer is determined in accordance with Newton's formula. The use of perturbation theory in the case of a thin blowing layer has made it possible to obtain limit problems for different flow regions, and the analytic solution and subsequent asymptotic matching of these problems yield the form of the contact surface and the distribution of the pressure on the body. It is shown that the drag of the body depends nonmonotonically on the flow rate of the blown gas. The optimal blowing parameters and the corresponding minimal drag are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 154–166, January–February, 1982.I thank V. A. Levin for interest in the work and valuable discussions.     

Effect of Localization of Pulsed Energy Addition on the Wave Drag of an Airfoil in a Transonic Flow

The possibility of controlling the aerodynamic characteristics of airfoils with the help of local pulsed-periodic energy addition into the flow near the airfoil contour at transonic flight regimes is considered. By means of the numerical solution of two-dimensional unsteady equations of gas dynamics, changes in the flow structure and wave drag of a symmetric airfoil due to changes in localization and shape of energy-addition zones are examined. It is shown that the considered method of controlling airfoil characteristics in transonic flow regimes is rather promising. For a zero angle of attack, the greatest decrease in wave drag is obtained with energy addition at the trailing edge of the airfoil.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 60–67, September–October, 2005.     

Determining the form of a body resulting in minimum heat flux,with laminar flow in the boundary layer

The exact solution of the problem of determining the optimal body shape for which the total thermal flux will be minimal for high supersonic flow about the body involves both computational and theoretical difficulties. Therefore, at the present time wide use is made of the inverse method, based on comparing the thermal fluxes for bodies of various specified form [1, 2]. The results of such calculations cannot always replace the solution of the direct variational problem. Therefore it is advisable to consider the direct variational problem of determining the form of a body with minimal thermal flux by using the approximate Newton formula for finding the gasdynamic parameters at the edge of the boundary layer. This approach has been used in finding the form of the body of minimal drag in an ideal fluid [3–5] arid with account for friction [6], and also for determining the form of a thin two-dimensional profile with minimal thermal flux for given aerodynamic characteristics [7].     

Natural Low-Frequency Riblets

A turbulent flow at modest Reynolds numbers (26,000–32,000, as defined in the text) over flat erodible sandy beds was investigated. The beds were composed of sand grains of uniform size with a diameter smaller than the viscous sublayer of the flow. When a modest sand transport (1.2 × 10-3–8.5 × 10-3 g/s·cm) was present, the near-wall coherent structures of the flow produced a bed form with a streaky longitudinal pattern. This pattern had a lateral spacing of about 100 viscous units and had less drag than a flat wall.The mechanism causing the drag reduction is due to a low-frequency riblet pattern which allows the flow to sustain higher wall shear stresses without additional erosion. That means during the riblet regime the sediment transport remains quasi-constant.The formation of the sand streaks as well as the drag reducing mechanism are discussed.     

Numerical study of the entry of a plate and a disk into a compressible fluid

The dependences of the drag force on the time and the Mach number are found, as also are pressure distribution, and the shape of the free surface. It is shown that with the passage of time the drag force rapidly approaches its asymptotic value, which corresponds to flow around a body by a compressible fluid in accordance with Kirchhoff's scheme. It is also shown that with increasing Mach number the dimensions of the cavity decrease, the unsteady cavity always being narrower than the Kirchhoff cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–107, March–April, 1985.     

Influence of real gas properties on the integral aerodynamic coefficients

The results of mathematically modeling axisymmetric hypersonic flow past an ellipsoid are presented. The calculation data are obtained on the basis of a numerical solution of the complete Navier—Stokes equations using a finite-difference method. The investigation is carried out for the problem of laminar flow over the windward side of the body. The effect of the elongation of the ellipsoid on the total heat flux and the viscous and pressure drag is considered. Results are obtained for three different gas (air) models: the perfect gas, chemical equilibrium and chemical nonequilibrium models. For the latter model various sets of catalyticity coefficients are considered. The effect of the real properties of air on the integral aerodynamic characteristics of ellipsoids with different elongations is analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 156–164, July–August, 1992.     

Simulation of the aerodynamic drag of three-dimensional star-shaped bodies at hypersonic velocities

The domain of the parameters in which the aerodynamic drag of hypersonic pyramidal bodies, whose wave component is calculated within the framework of conical flows with the boundary layer displacement thickness taken into account, agrees satisfactorily with the experimental data is found. The calculation model is also applicable in the region of minimum aerodynamic drag of star-shaped bodies in the class of conical bodies equivalent in length and mid-sectional area.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 69–79, September–October, 1996.     

Investigation of the shape of the transverse contour of a conical three-dimensional body of minimal drag moving in rarefied gas

The shape of bodies of minimal drag moving in rarefied gas in the entire range of flight heights is investigated at present on the basis of the use of local interaction models [1]. Corresponding theoretical investigations have been published in [2, 3] for the case of bodies of rotation and in [4] for three-dimensional winged bodies, and detailed numerical investigations have been carried out in [2, 5–7]. In the present paper, an analytic investigation is carried out for the purpose of determining the optimal shape of three-dimensional bodies with minimal drag in an intermediate region of flight heights in rarefied gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 112–117, September–October, 1986.     

On turbulent separated flow past a truncated body of revolution

Results of testing a series of truncated bodies of revolution with convergent afterbodies in a hydrodynamic tunnel are presented. It is shown that the base pressure can be substantially raised and hence the total drag reduced by varying the shape and convergence of the afterbodies. This effect is caused by intense reverse jets formed as a result of the collision of flow particles moving toward the axis of symmetry.The turbulent flow past the bodies is calculated using the method of viscous-inviscid interaction. The formulas derived for the base pressure and drag coefficients agree satisfactorily with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–55, November–December, 1996.     

Optimal lifting surfaces of complicated-geometry wings at supersonic flight velocities

Infinitely thin wings weakly perturbing a supersonic flow of perfect gas are investigated. The flow problem is solved in a linear formulation [1]. The shape of the wing in plan and the Mach number M of the oncoming flow are specified. The optimal wing surface is determined as a result of finding the function of the local angles of attack _M(x, z) which ensures a minimum of the drag coefficient c_x when there are limitations in the form of equalities on the lift coefficient c_y and the pitching moment m_z. A separationless flow regime is realized on the optimal wing for the given number M, and its subsonic leading edge does not experience a load [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 154–160, November–December, 1985.     

Hydrodynamic drag of an ellipsoidal droplet at small Reynolds numbers

At present, there is an absence of the accurate data on the influence of the shape of a droplet on its hydrodynamic drag and mass transfer without which the design of mass transfer apparatus is impossible [1–3]. Most often it is assumed that the drag of an ellipsoidal liquid droplet as it moves along the axis of symmetry is determined by the product of the drag of a spherical liquid droplet and a coefficient which takes into account the shape and is determined from the drag of a solid ellipsoid for which the exact solutions are known. It is shown below that this assumption is not always valid.Translated from Tzvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–8, May–June, 1987.     

Drag of a blocked fluidized bed

The characteristics of a blocked fluidized bed are presented. The drag of the first and second layers of disks forming the solid phase of the bed and the static pressure at any depth of the structurally inhomogeneous bed as a function of the characteristic parameters were obtained from experimental data. The relation between the density of arrangement of the drag elements in the bed and the stability of its specific drag depthwise is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1977.In conclusion, the author thanks A. A. Barmin for editing the articel and valuable methodological advice.     

Laminar boundary layer on the moving surface of a rankine oval

A numerical investigation has been made of the laminar boundary layer that arises on the moving surface of a cylindrical body (Rankine oval with relative elongation 4) that moves with constant velocity in an incompressible fluid. The distributions of the frictional stress on the surface of the cylinder for different velocities of the wall motion are found. Numerical integration was employed to determine the work needed to overcome the frictional drag, the pressure, and also the work expended on the motion of the moving surface of the body in the case of constant velocity. In the presence of a separation region the drag forces are calculated under the assumption that in the separation region the pressure and the frictional stress on the wall are constant and equal to the corresponding values at the singular point of the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 3, pp. 171–174, May–June, 1984.I thank G. G. Chernyi for constant interest in the work and discussing the results.     

Hydrodynamic drag of the dolphin

An experimental study of the hydrodynamic drag of the dolphin is reported. The results are compared with calculations for the steady-state motion of a similar rigid model, and previously reported data on the hydrodynamic drag of the dolphin are refined. The drag turns out to be much less than the calculated drag, even when the initial laminar region is taken into account. Several conclusions can be drawn from the results: 1) The experimental results show quite reliably that hydrodynamically the dolphin is in fact a unique phenomenon. 2) The phenomenon is explained in terms of Gray's paradox for the dolphin. 3) The mechanism by which the dolphin's drag is reduced is apparently controllable. 4) Flow around the dolphin can be assumed to be laminar undetached flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 160–164, March–April, 1971.