Outer Inviscid Flow Research Articles

Comment on 'Rise of total pressure in frictional flow'

(7) with r — ^/(x2 + z2) the distance from the symmetry axis. These results can now be used for the calculation of the viscous effect on the stagnation pressure for, respectively, a circular cylinder and a sphere placed in a parallel flow, for the limit of large Reynolds number.5 In that case the viscous layer thickness is small with respect to the body radius, and the local stagnation area can be approximated with the corresponding stagnation flow. Taking the inviscid outer flow distribution from potential theory, we find Cylinder:

A theory of three-dimensional interfacial vorticity dynamics

A three-dimensional theory of vorticity dynamics on an incompressible viscous and immiscible fluid–fluid interface, or interfacial vorticity dynamics for short, is presented as a counterpart of the vorticity dynamics on an arbitrarily curved rigid wall [J. Fluid Mech. 254, 183 (1993)]. General formulas with arbitrary Reynolds numbers Re are derived for determining (1) how much vorticity exists on an interface S, (2) how much vorticity is created from S and sent into the fluid per unit area in per unit time, and (3) the force and moment acted on a closed interface by the created vorticity thereon. The common feature and fundamental difference between interfacial vorticity dynamics and its rigid-wall counterpart are analyzed. In particular, on a free surface, the primary driving mechanism of vorticity creation is the balance between the shear stress (measured by tangent vorticity) and the tangent components of the surface-deformation stress alone, which results in a weak creation rate of O (Re−1/2) at large Re. Therefore, the exact form of the theory with its full complexity is of importance mainly at low Reynolds numbers, especially in understanding the small-scale coherent structures of interfacial turbulence. The vorticity creation rate at high-Re approximations, including an interfacial boundary layer of finite thickness and the limit of Re→∞ (the so-called Euler limit), is also studied, both allowing for a rotational inviscid outer flow. While for the former this leads to a generalization of Lundgren’s theory [in Mathematic Aspects of Vortex Dynamics, edited by R. E. Caflish (SIAM, Philadelphia, PA, 1989), pp. 68–79] and amounts to solving a linear boundary-layer problem, for the latter the creation rate can be directly obtained from an inviscid solution, leading to a dynamic evolution equation of interfacial vortex sheet. In three dimensions, a vortex sheet may bifurcate into a normal vorticity field, upon which the dependence of the sheet velocity is determined. A few examples are examined to illustrate different aspects and approximation levels of the general theory.

Method of solution of the hypersonic boundary layer equations in the case of strong viscous-inviscid interaction

A method of solving the boundary layer equations is developed taking into account the strong interaction between the boundary layer and the outer hypersonic inviscid flow. The method is aimed at solving problems whose salient feature is the possible upstream propagation of disturbances over distances comparable with the body length. The procedure for fitting a self-consistent contour of the “effective” body using an artificially formulated boundary value problem for an ordinary second-order differential equation, which lies at the basis of the method, is considered in detail. The method is applied to the problem of flow around a flat plate with roughness in the form of an embankment or a trench; the calculated results are presented.

The automatic control of boundary-layer transition ? Experiments and computation

In recent years there has been an increasing interest in the control of boundary-layer transition through the use of wall suction. In the current work suction is provided through one or more suction panels situated close to the leading edge of a plate. Experiments show that boundary-layer pressure fluctuation measurements can be used to identify the position of transition. Transition can be maintained at a desired location with minimum power consumption by employing an automatic adaptive feedback control loop which regulates the suction flow rates of two independent suction panels. This can be expressed as a constrained optimization problem. To allow the suction flow rates to be updated, a modified least mean squares algorithm is used within the control loop. Experimental measurements show that the control algorithm allows fast and stable convergence towards the optimum suction distribution for a double suction panel configuration. Numerical simulations have also been performed. The two-dimensional boundary layer was calculated allowing the viscous boundary layer to interact with the inviscid outer flow. Following linear stability theory the spatial growth rates are calculated by solving an Orr-Sommerfeld type eigenvalue problem, with the streamwise location of transition predicted via theeN -method. Applying the same optimization strategy as in the experiments, good qualitative agreement between computations and experiments was found. The optimization algorithm has been applied to computer models where the relation between suction flow rates and transition location is described by an empirical analytical function. This shows that the controller can in principle be applied to systems with more than two suction panels.

The Influence of Viscous Effects on the Motion of a Body Floating in Waves

This paper describes a method for calculating the forces experienced by a body floating in waves, including those due to vortex shedding from its surface. The method uses a purely theoretical approach, incorporating viscous forces, for calculating the motions of the body in the frequency domain. It involves the matching of an outer inviscid flow with the local flow in the regions of flow separation on the body, which must be well defined. The outer flow is computed by a three-dimensional singularity distribution technique and the inner flow by the discrete vortex method. The technique has been applied to the prediction of the motion response of barges floating in waves. The results compare favorably with experimental data.

Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 CB = 0.6 Ship Model—Part 1: Froude Numbers 0.16 and 0.316

Part 1 of this two-part paper presents results from a towing-tank experiment conducted in order to explicate the influence of wavemaking by a surface-piercing body on its boundary layer and wake and provide detailed documentation of the complete flow field appropriate for validating computational methods. Mean-velocity and pressure field measurements were performed for Froude numbers 0.16 and 0.316 for a 3.048 m Series 60 CB = 0.6 hull form at numerous stations from the bow to the stern and into the near wake. For Froude number = 0.316, free-surface effects are very significant, whereas for Froude number = 0.16, they are negligible, except near the bow, such that comparison of the results enables the identification of the salient features of the wave-induced effects. Wave profiles and local and global elevations were also measured. In addition, resistance tests were conducted. The experimental equipment and procedures are described and the results are discussed to point out the essential differences between the flows at low and high Froude number. On the forebody, the differences are primarily in the outer (inviscid) flow, except at the bow, whereas on the afterbody and in the near wake, both the inner (viscous) and outer flows are altered. The results are discussed to assess the nature of the interaction between wavemaking and the boundary layer and wake. Most of the interaction can be explicated as a result of the wave elevations, wave-induced pressure gradients, and the displacement effects of the boundary layer. Part 2 concerns scale effects on near-field wave patterns and comparisons between the experimental results and inviscid theory.

A numerical study of the laminar incompressible flow over a 6:1 prolate spheroid at 10° incidence

AbstractThe steady incompressible laminar flow field over a 6:1 prolate spheroid at 10° incidence and a Reynolds number of 1·6 × 106 is investigated numerically by solving a reduced set of the Navier‐Stokes equations. The present study moves one step beyond the boundary layer approximation by relaxing the requirement of an imposed pressure field to permit the calculation of both attached and longitudinal vortical flow fields. The results shed light on the flow properties over slender bodies at intermediate incidence. The longitudinal vortex is found to be weak relative to vortex‐dominated flows. Nevertheless, it has pronounced effects on the flow near the surface and on global features of the flow field. A displacement velocity which describes the effect of the vortical flow on the outer inviscid flow is defined. The line on the spheroid where the displacement velocity vanishes closely follows the projection of the vortex centreline on the surface of the spheroid. It is demonstrated numerically that the convergence of the skin friction lines is not a unique criterion for identifying a vortex flow.

Calculation of three-dimensional boundary layer on the leeward side of a delta wing of finite length in the hypersonic flow in viscous interaction regime

UDC 533.6.011.55:532.526.2-3 in the undisturbed flow, the length of the body, and the viscosity coefficient at the stagnation temperature. However, in the neighborhood of the apex of the wing and its leading edges a strong viscous interaction regime is often realized, if the hypersonic interaction parameter, calculated along the length of the region in question, Xx = M~ Re-1/2 ~> 1. The flow in the hypersonic boundary layer on a delta wing of finite length at an angle of attack was considered in (2) under conditions of strong interaction (X = oo) over the entire wing. It was shown that on the leeward side of the wing the displacement thickness increases considerably with increase in the angle of attack and that this is consistent with the experimental data (3). In (4) in relation to flow over delta wings in the strong viscous interaction regime at an angle of attack c~ ~ = (s/Reo) 1/4 (s is the wing aspect ratio) it was shown that on the leeward side in the neighborhood of the apex of the delta wing back flow transverse velocities develop in the boundary layer. The assumption that the strong viscous interaction regime is realized on the leeward side of a delta wing requires the satisfaction of the condition that M2(O~e/OX ~ - ce~ 2 ~> 1 (5), where ~e is the displacement thici~ess. However, this condition is not always satisfied. Moreover, in individual regions on the leeward side of the wing it is possible to have situations in which OYe/OX ~ < 0 or O~/Ox ~ - e~ ~ < 0 and local expansion conditions are realized. Hypersonic flow over triangular bodies at an angle of attack in the viscous interaction regime is considered. The results of calculating the parameters of the three-dimensional boundary layer on the leeward side of a wing and its aerodynamic characteristics are presented. 1. It is proposed to consider hypersonic viscous gas flow over a thin delta wing of finite length L at a small angle of attack a~ - Re~ -1/4. The coordinate system with origin at the apex of the delta wing is shown in Fig. 1. The x ~ axis is directed along the axis of symmetry of the wing, the z ~ axis along the span, and the y~ axis along the normal to the surface. The components W ~ of the velocity vector u ~ v ~ , are directed along x ~ , y~ z ~ respectively, and the aspect ratio of the wing s tan 8, where /3 is the apex half-angle. It is assumed that in the flow over the wing viscous interaction between the three-dimensional boundary layer and the outer inviscid flow is realized. Solving the complete boundary-value problem involves taking into account the flow in the wake formed behind the wing (6). However, in the present study, in order not to have to consider this flow, the boundary condition is imposed on the trailing edge of the delta wing. Since it is proposed to consider the flow over a noncold plate (gw ~ 0) and the flow is subcritical, the series expansion of the solution in the neighborhood of the leading edge will contain an arbitrary function (6). Therefore in order to obtain a unique solution of the boundary-value problem on the trailing edge it is also necessary to assign a function, as which we will take the distribution of the pressure p~ In accordance with the estimates for the boundary layer in a hypersonic flow (5) we introduce the dimen~sionless variables x~ z~ y~ ~t,~ w~ v~174

Higher-order effects in boundary-layer premixed combustion

Higher-order effects in the boundary-layer theory are included in the present analysis to evaluate the interaction of the expansion process from the premixed combustion of reacting gases with the potential flow. Two cases are considered: (1) the premixed combustion process, which takes place above the flat plate; and (2) the premixed flame, which is generated in the wake of the flat plate. In both cases, the strong expansion process, when interacting with the outer inviscid flow, produces relatively strong pressure gradients, modifying the aerodynamic structure of the flow. 14 refs.

Calculation of laminar separation bubbles in the wake inflation/deflation regime

A new interacting boundary-layer method is developed for calculating two-dimensional incompressible laminar separation bubbles. The method relies on a completely consistent mixed boundary value treatment of the outer inviscid flow. For attached flows, the new method is found to converge as fast as the classical Cauchy-Hilbert method and to yield the same solutions. For separated flows, the new method is found to converge two to three times faster than a comparable Cauchy-Hilbert method. The physical asymptotic theories that underlie this new numerical method will be discussed, in so far as they motivate the method.

Separation and wake of pulsating laminar flow

This is an experimental study of laminar pulsating flow that passes through separation and is led to a separated region. Detailed information is obtained on all interconnected regions, namely, the approaching attached laminar boundary layer, the immediate neighbourhood of separation, the separated shear layer, the large-scale vortical region of the wake and the outer inviscid flow. The approaching flow is pulsated with frequencies that include the natural frequencies of the wake. Flow visualizations are contrasted with instantaneous and averaged digital data obtained via laser-Doppler velocimetry and hot-wire anemometry. The interaction of the disturbing and natural frequencies is studied. Results are com pared with recent theoretical advances. It is found that the amplitude of the imposed disturbance grows sharply in the neighbourhood of separation.

Free convection from a semi-infinite vertical surface bounded by a horizontal wall in a porous medium

A study of the steady free convection flow along a semi-infinite vertical flat plate which is placed in a saturated porous medium and at an arbitrary distance d above a horizontal wall is performed. The first- and second-order boundary layer equations and outer inviscid flow equations are studied to find the effects of large, but finite, values of the Rayleigh number. Results are obtained for the two cases in which the flat plate is (i) isothermal and (ii) uniform flux. It is found that the first-order boundary layer solutions overestimate the local Nusselt number except for the uniform flux plate for distances along the plate less than about 0.4 d.

Darcian free convective flow about an impermeable horizontal surface bounded by a vertical wall

The problem of steady free convection in a porous medium adjacent to a horizontal impermeable heated surface, with wall temperature T ̄ W = T ̄ 0 + A x ̄ λ (0 ⩽ λ < 2) for x ̄ ⩾ 0 and T ̄ = T 0 for − d ⩽ x ̄ < 0 and a vertical wall with temperature T ̄ W = T ̄ 0 for x ̄ = − d, y ̄ > 0 , is investigated by the method of matched asymptotic expansions. The first—and second—order boundary layer equations and the outer inviscid flow equations are studied to find the effects of large, but finite values of the Rayleigh number. The small parameter in the perturbation series is found to be the inverse one-third power of the Rayleigh number. It is found that λ = 1 2 is a critical value and therefore results are presented in detail for the two cases of λ = 0 and 1. It is found that the first-order boundary layer solutions overestimate the local Nusselt number except when λ > 1 2 and only then for short distances along the heated surface. Further, the horizontal velocity of the fluid on the heated surface is overestimated by the first-order boundary layer theory except for λ < 1 2 and for x̄ sufficiently small.

Reply by Authors to S.M. Bogdonoff and S. Wang

W thank Prof. Bogdonoff and Mr. Wang for their comments on our work, although we disagree with them. We did not use selected test results, but based our conclusions on the data set obtained by F.K. Lu during his Master's thesis research. Subsequent work by Good win using fins at larger angles of attack also substantiated our conclusions. If conflicting data have been obtained at Princeton in the interim, we are unaware of them. We are willing to consider conflicting data if any exist. Bogdonoff and Wang question our assertion that conical flow symmetry exists in the outboard region of swept shock/boundary-layer interactions generated by fins. This assertion is supported not only by our data, but by a number of experimental and computational studies conducted in several countries. To our knowledge no form of far-field symmetry other than conical has ever been observed in fingenerated interactions. The matter of the spanwise growth of incoming boundarylayer thickness is admittedly not completely understood. The preponderance of known experiments has used rectangular flat plates that naturally generate boundary layers which grow in the spanwise direction at the upstream limit of a swept interaction. However, in one experiment, the flat-plate leading edge was intentionally swept back 45 deg in order to check such boundary-layer growth effects. The results showed the surface pressure footprints of 4and 10-deg fin interactions at Mach 3 to be invariant to leading-edge sweepback of the flat plate. We find this to be a plausible result; the scale of the incoming boundary layer apparently having exerted its influence only in the establishment of the inception zone, according to dimensional arguments. In further support of conical symmetry, note that the outer inviscid flow is naturally conical. The interaction requires an equilibrium between an outer conical flow and an inner viscous layer. This layer is extremely thin at high Reynolds numbers compared to the dimensions of the fin (as was the case in Ref. 1). Therefore, if the viscous layer refuses to respond to the symmetry of the outer flow, enormous gradients would have to be supported by this thin wall layer. Thus, it seems clear that swept shock/boundary-layer interactions must rapidly adopt the symmetry imposed by the outer flow. The length required for this to occur is the inception length referred to in our paper. This argument has been quantified in an order-of-magnitude analysis by Inger. As for the inception length itself, our description in Ref. 1 was admittedly preliminary. This was not the primary subject of our work, and at that time no one had studied inception lengths per se. More recent work clarifies this issue. It is clear that the inception zone merges gradually into the out-

Unsteady transonic flow over wings including inviscid/viscous interaction

A numerical procedure is presented for computing the unsteady transonic flowfield about swept wings undergoing time-dependent motions. The outer inviscid portion of the flow is assumed to be governed by the modified unsteady transonic small disturbance potential equation which is integrated in the time domain by means of an alternating direction implicit algorithm. Dominant effects of the shock/boundary-layer interaction are accdunted for by a simple empirically defined model. Viscous flow regions adjacent to the wing surface and in the trailing wake are described by a set of integral equations appropriate for compressible turbulent shear layers. The two-dimensional boundary-layer equations are applied quasistatically stripwise across the span. Coupling with the outer inviscid flow is implemented through use of the displacement thickness concept. Validity of the assumptions underlying the method is established by comparison with experimental data for the flow about a high aspect ratio transport wing having an advanced airfoil section.

Implicit Solution for the Shock-Layer Flow around General Bodies

The simplified forms of the Navier-Stokes equations are solved by an iterative implicit scheme along the body axis for angles of incidence up to 30 deg. The method of solution yields both the inviscid and viscous flows simultaneously and predicts bow shock at stations normal to the axis. The technique is developed from a comparatively simple linearization procedure and has an option to iterate between body stations for higher accuracy. Reliable procedures have been introduced to account for the effect of the axial pressure gradient and to adjust the step increment for a given convergence requirement. A blunt cone and a Shuttle Orbiter-like configuration were studied and good agreement is obtained with available laminar boundary-layer solutions and experimental data. ISCOUS flowfield solutions around a re-entry vehicle are needed for the prediction of heating distributions as well as for the interpretation of aerodynamic data to support the design and flight test processes. The traditional boundary- layer (BL) approach that decouples the viscous flow from the outer inviscid flow has been valuable in the past; however, for the problems of current interest, such as the Space Shuttle Orbiter and the maneuverable re-entry vehicle, a more ac- curate solution is being obtained from the coupled fluid dynamics equations. This paper presents a method suitable for complex configurations that may be described by a lofting technique used by the aerospace industry. The method is devised to solve the Euler equations for the inviscid region and the parabolic Navier-Stokes (PNS) equations for the viscous region along an identifiable main-flow direction. The PNS equations are obtained from the full NS equation by omitting the second-order and the mixed derivatives and by approximating the exact pressure gradient in the designated direction. The flowfield of interest is bounded between the shock, the body, an initial-data plane normal to the specified direction, and a downstream station parallel to the initial plane. Although the length scales associated with the flow phenomena differ greatly on that plane, highly nonuniform grid spacings near the wall are not desirable in practice, as the shock must be calculated accurately. Hence, a detailed shock- layer solution requires a large number of grid points and consequently long computation time. In return for its cost, this method does not have any of the known shortcomings associated with the BL analyses and offers new capabilities for predicting leeside flow and crossflow separation. For this reason, there has been an intensive research effort in this decade to solve for the viscous flow as an integral part of the flowfield.

Flow development in the vicinity of the sharp trailing edge on bodies impulsively set into motion

The initial development of the viscous flow in the vicinity of the sharp trailing edge of a symmetrical body, impulsively set into motion is studied within the framework of boundary-layer theory. For a blunt trailing edge there exists a similarity solution for the inviscid outer flow as has been pointed out by Proudman &amp; Johnson (1961). The present work shows that for sharp trailing edges, however, no such solution exists.For small or moderate trailing-edge angles, a moving singularity occurs in the solution fairly early in the flow development. The flow in the vicinity of this singularity exhibits the characteristics of unsteady separation. For large trailing-edge angles, the boundary layer becomes so thick that solutions must be terminated before there is any evidence of a singularity. For a cusped trailing edge, the solution is arbitrarily terminated to avoid large computational times and with the realization that the trailing edge flow must eventually be influenced by the leading edge at which time the present solutions cease to have meaning.

Non-uniform slot injection into a laminar boundary layer

Slot injection into a laminar boundary layer in both supersonic and subsonic flow is considered. The blowing rates are sufficiently large to provoke an interaction between the boundary layer and outer inviscid flow, and this interaction is accounted for by triple-deck theory. The non-uniform nature of the blowing velocity models the channel flow from which the injection takes place.

Supersonic Base Flow Problem in Presence of an Exhaust Jet

For the axisymmetric base flow problem in presence of a jet, a systematic study is carried out taking into account all essential physical parameters. Using the method of characteristi cs for the inviscid outer flow and the method of Korst for the turbulent shear layers, the influence of the approaching boundary layer is introduced by the concepts of equivalent bleed and origin shift, and the ONERA angular criterion is used for the recorepression process. A modification of the ONERA criterion for the two-stream recompression is proposed. The results are thoroughly discussed and compared with experiments, and estimations of the Reynolds number influence are presented.

Nonlinear analysis of laminar boundary layer flow over a periodic wavy surface—Part II: Long waves

The previous analysis of laminar, incompressible flow over a periodic wavy surface is extended to include wavelengths which are large compared with the mean boundary‐layer thickness. These larger wavelengths produce a stronger coupling between the outer inviscid flow and the viscous disturbance sublayer and, as a result, require the complete solution of the elliptic inviscid equations. The analysis is refined to allow for a net thickening of the mean boundary layer with increasing wave amplitude. Computer‐generated solutions are presented for sinusoidal waves of varying amplitude and wavelength, including cases with local flow separation.