Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

We establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Laval nozzle. The transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the exit at infinity, and the slip boundary condition on the nozzle boundary. Our results indicate that, if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow, there exists a solution that consists of a C ^1,α subsonic flow in the unbounded downstream region, converging to a uniform velocity state at infinity, and a C ^1,α multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow; the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow; and the shock is orthogonal to the nozzle boundary at every point of their intersection. In order to construct such a transonic flow, we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase, in which the equation is elliptic and the free boundary is a transonic shock. The free boundary conditions are determined by the Rankine–Hugoniot conditions along the shock. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain. We also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.     

Stabilization and evolution of parameters of a symmetric laminar flow in a plane channel with sudden expansion

A problem is formulated for computing the fields of parameters of a stationary laminar symmetric flow. A two-dimensional flow in a channel with a sudden change in the cross-sectional area is computed. The evolution of a three-dimensional perturbation inserted into the channel at the initial stage of computations is analyzed. It is demonstrated that the parameters of a two-dimensional flow in the channel at a Reynolds number Re = 50 become stabilized at a dimensionless time t > 20, whereas the steady state is reached under the same conditions at t ≈ 100. At a distance of approximately 10h (h is the channel width at the entrance), the flow becomes one-dimensional, but the streamwise component of the velocity vector remains a function of the streamwise coordinate owing to flow compressibility. __________ Translated from PrikladnayaMekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 35–42, January–February, 2007.     

Incompressible fluid flow simulations with flow rate as the sole information at synthetic inflow and outflow boundaries

In numerically simulating heat and mass transport processes in an unconfined domain involving synthetic open (inflow and/or outflow) boundaries, how to properly specify flow conditions at these boundaries can become a challenging issue. In this work, within the context of a pressure‐based finite volume method under an unstructured grid, a solution procedure without the need for explicit specification of flow profiles at any of these boundaries when simulating incompressible fluid flow is proposed and numerically examined. Within this methodology, the flow at any open boundary is not necessarily assumed to be unidirectional or fully developed; indeed, the sole information required is the mass flow rate crossing the boundary. As a result, one can select the specific region of interest to perform simulations, rather than having to artificially increase the flow domain so as to invoke fully developed flow at all open boundaries. This not only greatly reduces computational costs (both in terms of memory requirements and simulation run‐time) but provides the means to engage with flow problems, which otherwise cannot be solved with currently available methods for handling the flow conditions at open boundaries. The proposed methodology is demonstrated by simulating laminar flow of an incompressible fluid in a two‐dimensional planar channel with a 90° T‐branch, a known inflow rate, and flow splits for the two outflow channels. The results obtained by placing the entrance and the two exits at different locations show that the flow behavior predicted is completely unaffected by using a highly truncated domain. Copyright © 2015 John Wiley & Sons, Ltd.     

Peristaltic transport of a power-law fluid: Application to the ductus efferentes of the reproductive tract

The problem of peristaltic transport of a non-Newtonian (power-law) fluid in uniform and non-uniform two-dimensional channels has been investigated under zero Reynolds number with long wavelength approximation. A comparison of the results with those for a Newtonian fluid model shows that the magnitude of pressure rise, under a given set of conditions, is smaller in the case of the non-Newtonian fluid (power-law indexn < 1) at zero flow rate. Further, the pressure rise is smaller asn decreases from 1 at zero flow rate, is independent ofn at a certain value of flow rate and becomes greater if flow rate increases further. Also, at a given flow rate, an increase in wavelength leads to a decrease in pressure rise and increase in the influence of non-Newtonian behaviour. Pressure rise in the case of non-uniform geometry, is found to be much smaller than the corresponding value in the case of uniform geometry. Finally, the analysis is applied and compared with observed flow rates in the ductus efferentes of the male reproductive tract.     

渐缩平板间的缓变流

通过数值模拟方法探讨了渐缩平板间的不平行流动和缓变流的关系.分析了倾斜板的倾角对流动的影响.对采用EXP(C_ES)方式得到的数据,在板倾角为5.455°时不平行流动相对平行流动的偏差达1%,在13.26°时偏差达10%.这里取1%的偏差所对应的5.455°作为本文探讨的缓变流的临界角.最后讨论了数值模拟结果的合理性和可靠性.     

Three-dimensional numerical simulation of thermocapillary flow in a shallow cylindrical pool

In order to understand the characteristics of thermocapillary flow, we conducted a series of unsteady three-dimensional (3D) numerical simulations of thermocapillary flow for fluids with Prandtl number Pr = 0.011 and 1.0 in a shallow cylindrical pool with an azimuthal temperature non-uniform, an adiabatic solid bottom and free surface. The simulation results indicate that thermocapillary flow is steady 3D flow at the small temperature non-uniform. When temperature non-uniform exceeds some critical value, the flow will undergo a transition to 3D oscillatory flow, which is characterized by traveling oscillatory wave reversely along the flow direction on the free surface and diffusing to two sides at the same time. Further, the critical conditions for the flow transition are determined.     

A three-dimensional simulation of a steady approach flow past a circular cylinder at low Reynolds number

The three-dimensional (3D) unsteady viscous wake of a circular cylinder exposed to a steady approach flow is calculated using a fractional-step finite-difference/spectral-element method. The calculated flow fields at Reynolds numbers of 100 (2D) and 200 (3D) are examined in detail. The flow field at Re = 100 is 2D as expected, while the flow field at Re = 200 has distinct 3D features, with spanwise wavelengths of about 3.75 cylinder diameters. The calculated results produce drag and lift coefficients and Strouhal numbers that agree extremely well with the experimental values. These 3D values at Re = 200 are in better agreement with experimental values than the results of a 2D calculation at Re = 200, which is expected. © 1998 John Wiley & Sons, Ltd.     

Super-stable parallel flows of multiple visco-plastic fluids

We consider the stability of a multi-layer plane Poiseuille flow of two Bingham fluids. It is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone. This phenomenon of super-stability results only when the yield stress of the fluid next to the channel wall is larger than that of the fluid in the centre of the channel, which need not have a yield stress. Our result is in direct contrast to the stability of analogous flows of purely viscous generalised Newtonian fluids, for which short wavelength interfacial instabilities can be found at relatively low Reynolds numbers. The results imply the existence of parameter regimes where visco-plastic lubrication is possible, permitting transport of an inelastic generalised Newtonian fluid in the centre of a channel, lubricated at the walls by a visco-plastic fluid, travelling in a stable laminar flow at higher flow rates than would be possible for the single fluid alone.     

Measurements and high-speed visualizations of flow boiling of a dielectric fluid in a silicon microchannel heat sink

Experiments were conducted to investigate flow boiling heat transfer to a dielectric fluid in a silicon chip-integrated microchannel heat sink. Twenty-four microchannels, each 389 μm × 389 μm in cross-section, were fabricated into the 12.7 mm × 12.7 mm silicon substrate. High-speed visualizations (at 12,500 frames per second) were performed simultaneously with heat transfer and pressure drop measurements to investigate the physics of flow boiling in parallel microchannel arrays. At low heat fluxes, bubbly flow is dominant, with the bubbles coalescing to form vapor slugs as the heat flux is increased. At high heat fluxes, the flow regimes in the downstream portion of the microchannels are characteristic of alternating wispy-annular flow and churn flow, while flow reversal is observed in the upstream region near the microchannel inlet. Local heat transfer measurements, obtained at three flow rates ranging from 35 to 60 ml/min, show that at lower heat fluxes, the heat transfer coefficient increases with increasing heat flux. The heat transfer coefficient in fully developed boiling is seen to be independent of flow rate in this range. At higher heat fluxes (exceeding 542, 673, 730 kW/m², respectively, for flow rates of 35, 47 and 60 ml/min), this trend is reversed, and the heat transfer coefficient decreases with further increases in heat flux due to partial dryout in some of the microchannels. Heat fluxes at which fully developed boiling is achieved depend on the flow rate. The pressure drop in fully developed boiling increases with increasing heat flux and is independent of flow rate for the test conditions considered in this work.     

Contribution to the theory of thin-profile trailing edge separation

The conditions of nonsymmetric trailing edge flow with separation are investigated. Solutions of the equations for the interaction zone in the neighborhood of the trailing edge of a thin profile at an angle of attack of the order O(Re^–1/16) in the separated flow regime are constructed numerically. It is shown that for this zone a solution exists up to a certain angle of attack. In all the regimes the value of the friction on the upper surface at the very end of the trailing edge remains a positive quantity. The solution of the equations in the separated flow regimes is found to be nonunique. The flow over the leading edge is assumed to be unseparated, and the separation at the trailing edge, if present, is assumed to be localized in the interior of the boundary layer. The flow over a Kutta profile at zero angle of attack is taken as an example. In this case the satisfaction of the Chaplygin-Joukowsky condition at the trailing edge ensures smooth flow over both the trailing and leading edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–59, July–August, 1989.     

Turbulent flow in a circular separationless diffuser at Reynolds numbers smaller than 2000

The results of experimental and numerical investigation of flow in a circular conical diffuser with a small conicity angle ensuring separationless flow are presented. The measurements are carried out in an air flow with the Reynolds number Re₂ in the diffuser exit section ranging from 600 to 3000. A considerable effect of the channel expansion on the flow pattern is found to exist. It is shown that, as distinct from a tube, in which only laminar flow can be realized as steady for Re < 2000, in the exit section of a diffuser with the generator slope of 0.3° and a length equal to 70 entry diameters a developed turbulent flow is formed for Re₂ > 1000. For Re₂ > 1300 this flow is steady, that is, almost independent of the turbulence level at the entry, and is determined by the Reynolds number Re₂ in the exit section. For Re₂ ≈ 1000 the turbulent flow continuously goes over into a laminar flow. The flow parameters measured at the diffuser exit correspond to calculations in accordance with the threeequation turbulence model.     

Numerical simulation of cross-flow around four cylinders in an in-line square configuration

Successful numerical simulations can reveal important flow characteristics and information which are extremely difficult to obtain experimentally. Two- and three-dimensional (3-D) numerical simulations of cross-flow around four cylinders in an in-line square configuration are performed using a finite-volume method. For 2-D studies, the Reynolds numbers (Re) are chosen to be Re=100 and 200 and the spacing ratio L/D is set at 1.6, 2.5, 3.5, 4.0 and 5.0. For the 3-D investigation, the simulation is only performed at a Re=200, a spacing ratio L/D=4.0 and an aspect ratio H/D=16. The 2-D studies reveal three distinct flow patterns: (I) a stable shielding flow; (II) a wiggling shielding flow and (III) a vortex shedding flow. A transformation of the flow pattern from (I) to (II) at Re=100 will increase the amplitude of the maximum fluctuating pressure on the downstream cylinder surface by 4–12 times, while a transformation of the flow pattern from (II) to (III) will enhance the maximum fluctuating pressure amplitude by 2–3 times. There is a large discrepancy between 2-D simulation and flow visualization results at L/D=4.0 and Re=200. A probable cause could be the strong 3-D effect at the ends of the cylinder at low H/D. It was found that, for an in-line square configuration at critical L/D and when H/D is lower than a certain value, 3-D effects are very significant at the ends of the cylinders. In such cases, a time-consuming 3-D numerical simulation will have to be performed if full replication of the flow phenomenon were to be achieved.     

Non-intrusive measurements of convective heat transfer in smooth- and rough-wall microchannels: laminar flow

The convective heat transfer behavior of laminar flow through a smooth- and two rough-wall microchannels is investigated by performing non-intrusive and spatially resolved measurements of fluid temperature via two-color fluorescent thermometry under constant heat flux conditions at three of the four microchannel walls. Pressure-drop measurements reveal that the apparent friction factors for all surfaces agree well with established macroscale predictions for laminar flow through rectangular ducts with the onset of transition at Re > Re_cr = 1,800 for smooth-wall flow and deviation from laminar behavior at progressively lower Re with increasing surface roughness. The local Nu for smooth-wall flow agrees well with macroscale predictions in both the thermally developing and developed regimes. With increasing roughness, while an enhancement in local Nu is noted for flow in the thermally developing regime, no measurable influence is noted upon attainment of a thermally developed state. These observations are supported by the examination of temperature profiles across the microchannel at various axial positions and Re, which suggest that the thermal boundary layer may be regenerated locally by roughness in the thermal entrance region of the flow resulting in an increased axial distance (compared to smooth-wall behavior) at which thermally developed flow is attained in the presence of roughness. Finally, estimates of the bulk Nu indicate enhancement in convective heat transfer over the smooth-wall case for laminar flow at higher Re while the smooth-wall bulk Nu data are found to agree well with macroscale predictions.     

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10⁻⁶) and porous flow (low permeability Da < 10⁻⁶). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.     

Numerical study of hysteresis in annular swirling jets with a stepped‐conical nozzle

This study investigates the experimentally observed hysteresis in the mean flow field of an annular swirling jet with a stepped‐conical nozzle. The flow is simulated using the Reynolds‐averaged Navier–Stokes (RANS) approach for incompressible flow with a k–ε and a Reynolds stress transport (RSTM) turbulence model. Four different flow structures are observed depending on the swirl number: ‘closed jet flow’, ‘open jet flow low swirl’, ‘open jet flow high swirl’ and ‘coanda jet flow’. These flow patterns change with varying swirl number and hysteresis at low and intermediate swirl numbers is revealed when increasing and subsequently decreasing the swirl. The influence of the inlet velocity profile on the transitional swirl numbers is investigated. When comparing computational fluid dynamics with experiments, the results show that both turbulence models predict the four different flow structures and the associated hysteresis and multiple solutions at low and intermediate swirl numbers. Therefore, a good agreement exists between experiments and numerics. Copyright © 2006 John Wiley & Sons, Ltd.     

Validation of a novel very large eddy simulation method for simulation of turbulent separated flow

The paper describes the validation of a newly developed very LES (VLES) method for the simulation of turbulent separated flow. The new VLES method is a unified simulation approach that can change seamlessly from Reynolds‐averaged Navier–Stokes to DNS depending on the numerical resolution. Four complex test cases are selected to validate the performance of the new method, that is, the flow past a square cylinder at Re = 3000 confined in a channel (with a blockage ratio of 20%), the turbulent flow over a circular cylinder at Re = 3900 as well as Re = 140,000, and a turbulent backward‐facing step flow with a thick incoming boundary layer at Re = 40,000. The simulation results are compared with available experimental, LES, and detached eddy simulation‐type results. The new VLES model performs well overall, and the predictions are satisfactory compared with previous experimental and numerical results. It is observed that the new VLES method is quite efficient for the turbulent flow simulations; that is, good predictions can be obtained using a quite coarse mesh compared with the previous LES method. Discussions of the implementation of the present VLES modeling are also conducted on the basis of the simulations of turbulent channel flow up to high Reynolds number of Re_τ = 4000. The efficiency of the present VLES modeling is also observed in the channel flow simulation. From a practical point of view, this new method has considerable potential for more complex turbulent flow simulations at relative high Reynolds numbers. Copyright © 2013 John Wiley & Sons, Ltd.     

Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry

An analysis has been performed to study the unsteady laminar compressible boundary layer governing the hypersonic flow over a circular cone at an angle of attack near a plane of symmetry with either inflow or outflow in the presence of suction. The flow is assumed to be steady at time t=0 and at t>0 it becomes unsteady due to the time-dependent free stream velocity which varies arbitrarily with time. The nonlinear coupled parabolic partial differential equations under boundary layer approximations have been solved by using an implicit finite-difference method. It is found that suction plays an important role in stabilising the fluid motion and in obtaining unique solution of the problem. The effect of the cross flow parameter is found to be more pronounced on the cross flow surface shear stress than on the streamwise surface shear stress and surface heat transfer. Beyond a certain value of the cross flow parameter overshoot in the cross flow velocity occurs and the magnitude of this overshoot increases with the cross flow parameter. The time variation of the streamwise surface shear stress is more significant than that of the cross flow surface shear stress and surface heat transfer. The suction and the total enthalpy at the wall exert strong influence on the streamwise and cross flow surface shear stresses and the surface heat transfer except that the effect of suction on the cross flow surface shear stress is small.     

Unsteady Laminar to Turbulent Flow in a Spacer-Filled Channel

A combined numerical and experimental investigation has been carried out to study the flow behaviour in a spacer-filled channel, representative of those used in spiral-wound membrane modules. Direct numerical simulation and particle image velocimetry were used to investigate the fluid flow characteristics inside a 2 × 2 cell at Reynolds numbers that range between 100 and 1000. It was found that the flow in this geometry moves parallel to and also rotates between the spacer filaments and that the rate of rotation increases with Reynolds number. The flow mechanisms, transition process and onset of turbulence in a spacer-filled channel are investigated including the use of the velocity spectra at different Reynolds numbers. It is found that the flow is steady for Re < 200 and oscillatory at Re ～ 250 and increasingly unsteady with further increases in Re before the onset of turbulent flow at Re ～ 1000.     

Bluff-body drag reduction by passive ventilation

The drag of a sphere at highRe can be reduced to more than half its value by passive ventilation from the stagnation region to the base. Simultaneously, the flow field around the base is stabilized and made symmetric, leading to reduction of unsteady aerodynamic forces. At highRe, the vent flow breaks through the dead water region associated with the near wake and aerodynamically streamlines the base. The streamlining is done by virtue of a base-vortex-ring beyond the point of turbulent boundary layer separation. A mean flow model for the flow around the vented sphere is proposed.Smoke flow visualized on a laser light screen placed at two diameters behind the base of the sphere shows the effectiveness of the method in suppressing the flow oscillations.The drag reduction achieved is very sensitive to the quality of the external surface and relatively insensitive to disturbances in the internal flow. Surface roughness or boundary layer tripping wire on the external flow can completely offset the benefit obtained.     

Aspect ratio effects on three-dimensional incompressible flow in a two-sided non-facing lid-driven parallelepiped cavity

A numerical study of the three-dimensional fluid flow has been carried out to determine the effects of the transverse aspect ratio, Ay, on the flow structure in two-sided non-facing lid-driven cavities. The flow is complex, unstable and can undergo bifurcation. The numerical method is based on the finite volume method and multigrid acceleration. Computations have been investigated for several Reynolds numbers and various aspect ratio values. At a fixed Reynolds number, Re=500, the three-dimensional flow characteristics are analyzed considering four transverse aspect ratios, Ay=1,0.75,0.5 and 0.25. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. An analysis of the flow evolution shows that, at Ay=0.75, the flow bifurcates to a periodic regime at (Re=600) with a frequency f=0.093 less than the predicted value in the cubical cavity. A correlation is established when Ay=0.5 and gives the critical Reynolds number value. At Ay=0.25, the periodic regime occurs at high Re value beyond 3500, after which the flow becomes chaotic. It is shown that, when increasing Ay over the unit, the flow in the cavity exhibits a complex behavior. The kinetic energy transmission from the driven walls to the cavity center is reduced at low Ay values.