Two-dimensional turbulence in square and circular domains with no-slip walls

Several fascinating phenomena observed for 2D turbulence in bounded domains are discussed. The first part of this paper concerns a short overview of the non-trivial behaviour of freely evolving 2D turbulence in square domains with no-slip boundaries. In particular, the Reynolds number dependence of, and the influence of the initial conditions on spontaneous spin-up of the flow, which is characterised by a sudden increase of the absolute value of the angular momentum of the flow, is investigated in more detail. In a second set-up we have investigated forced 2D turbulence in circular containers with no-slip walls. A comparison with the double periodic case reveals that domain-filling structures, always observed in the double periodic cases, are being prevented from emerging. Wall-generated, small-scale structures are continuously injected into the interior of the domain, destroying larger structures and maintaining the turbulent flow field.     

星载静电加速度计加转回路建模与分析

新型等效原理实验通过检验两个相同材料但自旋运动有显著差异的宏观物体的自由落体运动来验证等效原理可能存在的破坏。根据新型等效原理空间实验的科学目标,提出了一种星载差分静电加速度计方案,设计了基于可变电容式电机原理的静电加转回路;在分析了作用于转子上的静电加转力矩、残余气体阻尼及磁场阻尼的基础上,建立了转子加转回路的动力学模型并进行了仿真分析;仿真结果表明,启动过程使转子达到目标转速(10 000 rpm)的启动时间为9.8天。静电加转方法可在电极筒上同时配置加转电极与悬浮电极,结构紧凑,同时避免了传统的异步电机加转产生的电磁干扰。     

Thermal nonequilibrium,non-darcian forced convection in a channel filled with a fluid saturated porous medium—A perturbation solution

This paper concentrates on the analysis of the thermal nonequilibrium effects during forced convection in a parallel-plate channel filled with a fluid saturated porous medium. The flow in a channel is described by the Brinkman-Forchheimer-extended Darcy equation and the thermal nonequilibrium effects are accounted for by utilizing the two energy equations model. Applying the perturbation technique, an analytical solution of the problem is obtained. It is established that the temperature difference between the fluid and solid phases for the steady fully developed flow is proportional to the ratio of the flow velocity to the mean velocity. This results in a local thermal equilibrium at the walls of the channel if the Brinkman term which allows for the no-slip boundary condition at the walls is included into the momentum equation.     

Gas-particle two-phase turbulent flow in a vertical duct

Two-phase gas-phase turbulent flows at various loadings between the two vertical parallel plates are analyzed. A thermodynamically consistent turbulent two-phase flow model that accounts for the phase fluctuation energy transport and interaction is used. The governing equation of the gas-phase is upgraded to a two-equation low Reynolds number turbulence closure model that can be integrated directly to the wall. A no-slip boundary condition for the gas-phase and slip-boundary condition for the particulate phase are used. The computational model is first applied to dilute gas-particle turbulent flow between two parallel vertical walls. The predicted mean velocity and turbulence intensity profiles are compared with the experimental data of Tsuji et al. (1984) for vertical pipe flows, and good agreement is observed. Examples of additional flow properties such as the phasic fluctuation energy, phasic fluctuation energy production and dissipation, as well as interaction momentum and energy supply terms are also presented and discussed.

Applications to the relatively dense gas-particle turbulent flows in a vertical channel are also studied. The model predictions are compared with the experimental data of Miller & Gidaspow and reasonable agreement is observed. It is shown that flow behavior is strongly affected by the phasic fluctuation energy, and the momentum and energy transfer between the particulate and the fluid constituents.     

In-cylinder swirl formation process in a four-valve diesel engine

An experimental study is presented on the steady flow in a four-valve diesel engine rig by using hot-wire anemometry. Through the analysis of the in-cylinder three-dimensional flow field of a four-valve diesel engine, the in-cylinder swirl formation process at various valve lift was investigated. A new criterion is proposed for predicting the emergence of a stable-swirl formation interface, based on swirl angular momentum flux. A stable-swirl formation interface exists when the main swirl angular momentum flux is nearly equal to the in-cylinder air total angular momentum flux. Received: 13 March 2000/Accepted: 7 February 2001     

Flow instability in a curved porous channel formed by two concentric cylindrical surfaces

Stability of laminar flow in a curved channel formed by two concentric cylindrical surfaces is investigated. The channel is occupied by a fluid saturated porous medium; the flow in the channel is driven by a constant azimuthal pressure gradient. The momentum equation takes into account two drag terms: the Darcy term that describes friction between the fluid and the porous matrix, and the Brinkman term, which allows imposing the no-slip boundary condition at the channel walls. An analytical solution for the basic flow velocity is obtained. Numerical analysis is carried out using the collocation method to investigate the onset of instability leading to the development of a secondary motion in the form of toroidal vortices. The dependence of the critical Dean number on porosity and the channel width is analyzed.     

Locally similar solutions for hydromagnetic and thermal slip flow boundary layers over a flat plate with variable fluid properties and convective surface boundary condition

This paper presents heat transfer process in a two-dimensional steady hydromagnetic convective flow of an electrically conducting fluid over a flat plate with partial slip at the surface of the boundary subjected to the convective surface heat flux at the boundary. The analysis accounts for both temperature-dependent viscosity and temperature dependent thermal conductivity. The local similarity equations are derived and solved numerically using the Nachtsheim-Swigert iteration procedure. Results for the dimensionless velocity, temperature and ambient Prandtl number within the boundary layer are displayed graphically delineating the effect of various parameters characterizing the flow. The results show that momentum boundary layer thickness significantly depends on the surface convection parameter, Hartmann number and on the sign of the variable viscosity parameter. The results also show that plate surface temperature is higher when there is no slip at the plate compared to its presence. For both slip and no-slip cases surface temperature of the plate can be controlled by controlling the strength of the applied magnetic field. In modelling the thermal boundary layer flow with variable viscosity and variable thermal conductivity, the Prandtl number must be treated as a variable irrespective of flow conditions whether there is slip or no-slip at the boundary to obtain realistic results.     

Three-dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts

The laminar convective flow and heat transfer in a duct with a trapezoidal cross-sectional area are studied numerically. The governing equations are solved numerically by a finite volume formulation in complex three-dimensional geometries using co-located variables and Cartesian velocity components. Details of the numerical method are presented. The accuracy of the method was also established by comparing the calculated results with the analytical and numerical results available in the open literature. The Nusselt numbers are obtained for the boundary condition of a uniform wall temperature whereas the friction factors are calculated for no-slip conditions at the walls. The asymptotic values of the Nusselt numbers, friction factors. incremental pressure drops, axial velocity and momentum rate and kinetic energy correction factors approach the available fully developed values. Various geometrical dimensions of the cross-section are considered.     

The effect of an additional boundary condition on the plane creeping flow of a second-order fluid

A similarity solution is obtained for the two-dimensional creeping flow of a second-order fluid with non-parallel porous walls. The resulting ordinary differential equation is fifth order. Thus, an additional velocity boundary condition is needed, the other four being due to the usual no-slip conditions. Having chosen to prescribe the rate of shear at the wall, the problem is solved by a standard numerical routine. A singular perturbation analysis is developed for small values of the Deborah number. A type of boundary layer forms for which the viscous Newtonian case is the outer solution.     

Run-up and spin-up in a viscoelastic fluid. I.

Summary An incompressible viscoelastic fluid is contained between parallel rigid plates which at some instant of time are subjected to a translational velocity which is then held constant. The dependence on time of the resulting flow field in the fluid is calculated. Similar calculations are carried out when the fluid is contained in an infinitely long circular cylinder which is set in longitudinal motion or in rotation.It is seen that for a certain broad class of viscoelastic fluids the transfer of momentum, or angular momentum, from the boundary to the interior of the fluid takes place by a mechanism which is essentially diffusive in character. For another broad class of fluids, of which the Maxwellian fluid is a special case, the transfer of momentum results from the reflection back and forth of a velocity shock wave. These reflections take place at the boundaries in the case of runup between parallel plates and at the boundary and axis in the case of run-up, or spin-up, in a circular cylinder.With 4 figures     

Effects of slip condition on MHD stagnation-point flow over a power-law stretching sheet

The steady two-dimensional magnetohydrodynamic stagnation flow towards a nonlinear stretching surface is studied. The no-slip condition on the solid boundary is replaced with a partial slip condition. A scaling group transformation is used to get the invariants. Using the invariants, a third-order ordinary differential equation corresponding to the momentum is obtained. An analytical solution is obtained in a series form using a homotopy analysis method. Reliability and efficiency of series solutions are shown by the good agreement with numerical results presented in the literature. The effects of the slip parameter, the magnetic field parameter, the velocity ratio parameter, the suction velocity parameter, and the power law exponent on the flow are investigated. The results show that the velocity and shear stress profiles are greatly influenced by these parameters.     

Elementary properties of the enstrophy and strain fields in confined two-dimensional flows

The evolution of decaying two-dimensional turbulent flows in a bounded domain is considered. It is shown that the global enstrophy is always equal to the total squared strain field, when integrated over a domain with no-slip boundaries, despite the complex evolution of the flow and strong vortex-wall interactions. This property is also valid for a square domain with stress-free walls, and in general for any polygonal boundary. In contrast, the enstrophy is always greater than the squared strain field in a stress-free circular domain, and in general for any closed domain with negatively-signed curvature at all points of the boundary.     

Slip yield stress effects in start-up Newtonian Poiseuille flows

Analytical solutions are derived for various start-up Newtonian Poiseuille flows assuming that slip at the wall occurs when the wall shear stress exceeds a critical value, known as the slip yield stress. Two distinct regimes characterise the steady axisymmetric and planar flows, which are defined by a critical value of the pressure gradient. If the imposed pressure gradient is below this critical value, the classical no-slip, start-up solution holds. Otherwise, no-slip flow occurs only initially, for a finite time interval determined by a critical time, after which slip does occur. For the annular case, there is an additional intermediate (steady) flow regime where slip occurs only at the inner wall, and hence, there exist two critical values of the pressure gradient. If the applied pressure gradient exceeds both critical values, the velocity evolves initially with no-slip at both walls up to the first critical time, then with slip only along the inner wall up to the second critical time and finally with slip at both walls.     

Stokes Plane-Parallel Vortex Systems in Channels

Plane-parallel vortex systems in a viscous incompressible fluid in channels with parallel walls and in a corner with no-slip conditions on the walls are investigated on the basis of exact solutions of the biharmonic equation. It is found that separation zones and paired (joined) vortices are formed when the fluid flows through these channels and the fluid flow path between these vortex formations is traced. The flow structures are considered in the region of intrusion of the Poiseuille or Hamel flow into the zone of predominance of the intense vortex formations and for outflow of the fluid from this zone.     

The effects of the side walls on the flow of a second grade fluid in ducts with suction and injection

The effects of the side walls on the flow in ducts with suction and injection are examined. Three illustrative examples are given. The first example considers the effect of the side walls on the flow over a porous plate. The second example considers the flow between two parallel porous plates and the third example is devoted to the investigation of the flow in a rectangular duct with two porous walls. Exact solution of the governing equation using the no-slip boundary condition and an additional condition are obtained. The expression of the velocity, the volume flux and the vorticity are given. It is found that for large values of the cross-Reynolds number near the suction region the flow for a Newtonian fluid does not satisfy the boundary condition, but it does not behave in the same way for a second grade fluid. Three examples considered show that there are pronounced effects of the side walls on the flows of a second grade fluid in ducts with suction and injection.     

Cessation of Newtonian circular and plane Couette flows with wall slip and non-zero slip yield stress

We solve analytically the cessation flows of a Newtonian fluid in circular and plane Couette geometries assuming that wall slip occurs provided that the wall shear stress exceeds a critical threshold, the slip yield stress. In steady-state, slip occurs only beyond a critical value of the angular velocity of the rotating inner cylinder in circular Couette flow or of the speed of the moving upper plate in plane Couette flow. Hence, in cessation, the classical no-slip solution holds if the corresponding wall speed is below the critical value. Otherwise, slip occurs only initially along both walls. Beyond a first critical time, slip along the fixed wall ceases, and beyond a second critical time slip ceases also along the initially moving wall. Beyond this second critical time no slip is observed and the decay of the velocity is faster. The velocity decays exponentially in all regimes and the decay is reduced with slip. The effects of slip and the slip yield stress are discussed.     

Two-Dimensional Turbulence with Rigid Circular Walls

We report numerical computations of decaying two-dimensional Navier--Stokes turbulence inside a circular rigid boundary. We summarize previously reported calculations involving no-slip boundary conditions and present results with higher spatial resolution than achieved before (with, however, no qualitative changes in the observed behavior). We then report new results with stress-free boundary conditions (for a viscous fluid, but bounded by a perfectly slippery wall). The method used is spectral, involving expansions of the fields in orthonormal sets of functions which obey two boundary conditions (circular analogues of the Chandrasekhar–Reid functions). The computation takes place entirely in the spectral space. Large-scale Reynolds numbers are typically less than a thousand. Interest focuses on the role played by angular momentum, in determining the decay of the turbulence with no-slip boundary conditions, and the role of possible other ideal invariants in the stress-free case. Received 30 September 1996 and accepted 5 February 1997     

Influence of nonuniformity of supersonic flow with shock waves on the friction and heat transfer in planar channels

A numerical investigation has been made of the influence of nonuniformity of a supersonic flow with shock waves on the friction and heat transfer in planar channels. It is shown that for channels with relative length more than four diameters the frictional force applied to the inner walls of the channel and the total heat flux to the wall in the case of a nonuniform supersonic flow at the entrance correspond approximately to the corresponding quantities for the uniform flow that is equivalent to the nonuniform flow as regards the flow rate, energy, and momentum. The calculated values of the momentum loss coefficients for planar channels agree satisfactorily with the experimental data obtained by Ostras' and Penzin [1] for axisym-metric tubes with identical conditions at the entrance.