Numerical simulation of flow and heat transfer in radially rotating microchannels

Investigation of fluid flow and heat transfer in rotating microchannels is important for centrifugal microfluidics, which has emerged as an advanced technique in biomedical applications and chemical separations. The centrifugal force and Coriolis force, arising as a consequence of the microchannel rotation, change the flow pattern significantly from the symmetric profile of a non-rotating channel. Successful design of microfluidic devices in centrifugal microfluidics depends on effectively regulating these forces in rotating microchannels. In this work, we have numerically investigated the flow and heat transfer in rotating rectangular microchannel with continuum assumption. A pressure-based finite-volume technique with a staggered grid was applied to solve the steady incompressible Navier–Stokes and energy equations. It was observed that the effect of Coriolis force was determined by the value of the non-dimensional rotational Reynolds number (Re _ω ). By comparing the root mean square deviation of the axial velocity profiles with the approximate analytical results of purely centrifugal flow for different aspect ratios (AR = width/height), a critical rotational Reynolds number (Re _ω,cr) was computed. Above this value of (Re _ω,cr), the effect of secondary flow becomes dominant. For aspect ratios of 0.25, 0.5, 1.0, 2.0, 4.0 and 9.09, this critical rotational Reynolds number (Re _ω,cr) was found to be 14.0, 5.5, 3.8, 4.7, 6.5 and 10.0, respectively.     

Transition in a 2-D lid-driven cavity flow

Direct numerical simulations about the transition process from laminar to chaotic flow in square lid-driven cavity flows are considered in this paper. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasi-periodic, inverse period-doubling, period-doubling, and chaotic self-sustained flow regimes. The numerical experiments are conducted by solving the 2-D incompressible Navier-Stokes equations with increasing Reynolds numbers (Re). The spatial discretization consists of a seventh-order upwind-biased method for the convection term and a sixth-order central method for the diffusive term. The numerical experiments reveal that the first Hopf bifurcation takes place at Re equal to 7402±4%, and a consequent periodic flow with the frequency equal to 0.59 is obtained. As Re is increased to 10,300, a new fundamental frequency (FF) is added to the velocity spectrum and a quasi-periodic flow regime is reached. For slightly higher Re (10,325), the new FF disappears and the flow returns to a periodic regime. Furthermore, the flow experiences an inverse period doubling at 10,325 <Re< 10,700 and a period-doubling regime at 10,600 <Re< 10,900. Eventually, for flows with Re greater than 11,000, a scenario for the onset of chaotic flow is obtained. The transition processes are illustrated by increasing Re using time-velocity histories, Fourier power spectra, and the phase-space trajectories. In view of the conducted grid independent study, the values of the critical Re presented above are estimated to be accurate within ±4%.     

Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD)

The streamline-upwind/Petrov-Galerkin (SUPG) and pressure-stabilizing/Petrov-Galerkin (PSPG) methods are among the most popular stabilized formulations in finite element computation of flow problems. The discontinuity-capturing directional dissipation (DCDD) was first introduced as a complement to the SUPG and PSPG stabilizations for the computation of incompressible flows in the presence of sharp solution gradients. The DCDD stabilization takes effect where there is a sharp gradient in the velocity field and introduces dissipation in the direction of that gradient. The length scale used in defining the DCDD stabilization is based on the solution gradient. Here we describe how the DCDD stabilization, in combination with the SUPG and PSPG stabilizations, can be applied to computation of turbulent flows. We examine the similarity between the DCDD stabilization and a purely dissipative energy cascade model. To evaluate the performance of the DCDD stabilization, we compute as test problem a plane channel flow at friction Reynolds number Re_τ = 180.     

Cell pattern and stagnation ring of the flow driven by the counter-rotation in a fluid-filled cylinder

For the flow driven by the counter-rotation between the top and bottom endwalls in a fluid-filled cylinder, a stagnation ring can be observed on the slower rotating endwall in experiment. Its appearance corresponds to a two-cell flow pattern in the meridional plane, where a flow separation forms in the Ekman boundary layer. In this paper we numerically show that, in addition to the single-cell and two-cell patterns previously studied, there exist more complex cell patterns, namely, three-cell and merged-cell patterns, when the flow is driven under differently counter-rotating manner that is realized between the top and bottom endwalls as a whole against the sidewall. Such a counter-rotating flow makes the stagnation ring to appear simultaneously on both top and bottom endwalls rather than just on the slower rotating endwall. Moreover, the three-cell and merged-cell patterns, which are formed by a combination of the Ekman layer separation with the “vortex breakdown bubble”, are unique characteristics. The appearance of the cell pattern and stagnation ring is primarily decided by the counter rotation-rate ratio s, but is also affected by the Reynolds number Re and height-to-radius aspect ratio Λ, so a cell-pattern zone and a stagnation ring zone are proposed numerically as a function of s and Re for a given Λ.     

Enhancement of an industrial finite-volume code for large-eddy-type simulation of incompressible high Reynolds number flow using near-wall modelling

We present a validation strategy for enhancement of an unstructured industrial finite-volume solver designed for steady RANS problems for large-eddy-type simulation with near-wall modelling of incompressible high Reynolds number flow. Different parts of the projection-based discretisation are investigated to ensure LES capability of the numerical method. Turbulence model parameters are calibrated by using a minimisation of least-squares functionals for first and second order statistics of the basic benchmark problems decaying homogeneous turbulence and turbulent channel flow. Then the method is applied to the flow over a backward facing step at Re_h = 37,500. Of special interest is the role of the spatial and temporal discretisation error for low order schemes. For wall-bounded flows, present results confirm existing best practice guidelines for mesh design. For free-shear layers, a sensor to quantify the resolution quality of the LES based on the resolved turbulent kinetic energy is presented and applied to the flow over a backward facing step at Re_h = 37,500.     

Sudden expansions in circular microchannels: flow dynamics and pressure drop

Micro particle shadow velocimetry is used to study the flow of water through microcircular sudden expansions of ratios e = 1.51 and e = 1.96 for inlet Reynolds numbers Re _d < 120. Such flows give rise to annular vortices, trapped downstream of the expansions. The dependency of the vortex length on the Reynolds number Re _d and the expansion ratio e is experimentally investigated in this study. Additionally, the shape of the axisymmetric annular vortex is quantified based on the visualization results. These measurements favorably follow the trends reported for larger scales in the literature. Redevelopment of the confined jet to the fully developed Poiseuille flow downstream of the expansion is also studied quantitatively. Furthermore, the experimentally resolved velocities are used to calculate high resolution static pressure gradient distributions along the channel walls. These measurements are then integrated into the axisymmetric momentum and energy balance equations, for the flow downstream of the expansion, to obtain the irreversible pressure drop in this geometry. As expected, the measured pressure drop coefficients for the range of Reynolds numbers studied here do not match the predictions of the available empirical correlations, which are commonly based turbulent flow studies. However, these results are in excellent agreement with previous numerical calculations. The pressure drop coefficient is found to strongly depend on the inlet Reynolds number for Re _d < 50. Although no length-scale effect is observed for the range of channel diameters studied here, for Reynolds numbers Re _d < 50, which are typical in microchannel applications, complex nonlinear trends in the flow dynamics and pressure drop measurements are discovered and discussed in this work.     

Multiplicity of states for two-sided and four-sided lid driven cavity flows

Numerical simulations for incompressible flow in two-sided and four-sided lid driven cavities are reported in the present study. For the two-sided driven cavity, the upper wall is moved to the right and the left wall to the bottom with equal speeds. For the four-sided driven cavity, the upper wall is moved to the right, the lower wall to the left, while the left wall is moved downwards and the right wall upwards, with all four walls moving with equal speeds. At low Reynolds numbers, the resulting flow field is symmetric with respect to one of the cavity diagonals for the two-sided driven cavity, while it is symmetric with respect to both cavity diagonals for the four-sided driven cavity. At a critical Reynolds number of 1073 for the two-sided driven cavity and 129 for the four-sided driven cavity, the flow field bifurcates from a stable symmetric state to a stable asymmetric state. Three possible flow solutions exist above the critical Reynolds number, an unstable symmetric solution and two stable asymmetric solutions. All three possible solutions are recovered in the present study and flow bifurcation diagrams are constructed. Moreover, it is shown that the marching direction of the iterative solver determines which of the two asymmetric solutions is recovered.     

Effect of the Reynolds number on streamline bifurcations in a double-lid-driven cavity with free surfaces

The two-dimensional Navier-Stokes equations for a Newtonian fluid in the absence of body forces are considered in a rectangular double-lid-driven cavity with free surface side walls, the cavity aspect ratio A and three cases of the ratio (S=0,−1,1) of the upper to the lower lid speed. Using a finite element formulation with a mesh which is adaptively refined to facilitate the location of stagnation points, the effect of Reynolds numbers (Re) in the range [0,100] on the streamline patterns and their bifurcations is investigated as A is varied for each S. For Re→0 and each S as A is decreased, a sequence of pitchfork bifurcations at a stagnation point on x=0 is identified as seen in the work of Gaskell et al. [Proc Instn Mech Engrs Sci Part C 212 (1998) 387]. As Re increases for S=0 and decreasing A the stagnation point on x=0 disappears and away from x=0 cusp (saddle-node) bifurcations arise rather than the pitchfork bifurcation whereas for S=−1 and Re∈[0,100] the origin is always a stagnation point at which the same type of bifurcations arises.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Numerical simulation of two-dimensional turbulence in a plane channel

Two-dimensional flow of an incompressible viscous fluid in a plane channel is studied under fixed flux and supercritical Reynolds number, Re = 10⁴. For numerical simulation, and original algorithm, possessing good stability and accuracy properties, is used. Calculation of the flow over a large interval of time leads to a statistically steady-state condition of the flow and to stabilization of the averaged characteristics, e.g., profiles of averaged velocity, averaged gradient of pressure, energy, etc. The computations show that qualitatively proper characteristics of “two-dimensional” turbulence can be obtained by numerical simulation of the Navier-Stokes equations.     

On the use of several compact methods for the study of unsteady incompressible viscous flow round a circular cylinder

Fourth order accurate methods of mehrstellen type are compared to second order accurate methods for the solution of the unsteady incompressible Navier-Stokes equations in their vorticity stream function formulation. These methods are applied to the study of separated flow around a circular cylinder at several Reynolds numbers. The impulsively started cylinder at Re = 200 and 550, is considered without symmetry restrictions. The features illustrated include the bulge phenomenon at Re = 200, the occurrence of secondary vortices depending on the schemes used at Re = 550, and of twin secondary vortices at Re = 3000. The Karman vortex street is investigated at Re = 200 with a uniform flow in the far field and with superimposed motions of the cylinder. In this last case, a frequency analysis has allowed a critical examination of results pertaining to locked-in situations with respect to confinement effects.     

Numerical prediction of unsteady vortex shedding for large leading-edge roughness

A full two-dimensional Navier-Stokes algorithm is used to investigate unsteady, incompressible viscous flow past an airfoil leading edge with surface roughness that is characteristic of ice accretion. The roughness is added to the surface through the use of a Prandtl transposition and can generate both small-scale and large-scale roughness. The focus of the study is a detailed flow analysis of the unsteady velocity fluctuations and vortex shedding induced by the surface roughness. The results of this study are compared to experimental data on roughness-induced transition for the same roughness geometry. A comparison is made between “fluctuation intensity” values from the current algorithm to experimentally determined turbulence intensity values. The effects of the roughness Reynolds number, Re_k, are investigated and compared to experimental values of the critical roughness Reynolds number. The authors speculate that there may be a possible correlation between unsteady roughness-induced vortex shedding and the onset of experimentally measured transitional flow downstream of large-scale roughness.     

Identification of large coherent structures in supersonic axisymmetric wakes

Direct numerical simulation data of supersonic axisymmetric wakes are analysed for the existence of large coherent structures. Wakes at Ma=2.46 are considered with results being presented for cases at Reynolds numbers Re_D=30,000 and 100,000. Criteria for identification of coherent structures in free-shear flows found in the literature are compiled and discussed, and the role of compressibility is addressed. In particular, the ability and reliability of visualisation techniques intended for incompressible shear-flows to educe meaningful structures in supersonic wakes is scrutinised. It is shown that some of these methods retain their usefulness for identification of vortical structures as long as the swirling rate is larger than the local compression and expansion rates in the flow field. As a measure for the validity of this condition in a given flow the ‘vortex compressibility parameter’ is proposed which is derived here. Best ‘visibility’ of coherent structures is achieved by employing visualisation techniques and proper orthogonal decomposition in combination with the introduction of artificial perturbations (forcing of the wake). The existence of both helical and longitudinal structures in the shear layer and of hairpin-like structures in the developing wake is demonstrated. In addition, elongated tubes of streamwise vorticity are observed to emanate from the region of recirculating flow.     

Inertia effect on deformation of viscoelastic capsules in microscale flows

The deformation of capsules (i.e., cells, bacterial) in microscale flows plays an important role in biofluid flows such as blood flow in capillaries and cell manipulation in microfluidics. In previous studies on capsule deformation in microscale flows, the inertia effect was often assumed to be negligible and thus omitted. However, this assumption may not reflect real situations, as indicated by recent studies of inertial microfluidics. As such, we aimed to study the inertia effect on capsule deformation in microscale flows and to determine under which conditions this effect may be omitted. Using a collocated grid projection scheme, we developed a finite difference-front tracking method, and investigated the deformation of viscoelastic capsules in microscale flows for Reynolds number (Re) ranging from 0.01 to 10 as seen in vitro and in vivo. The results showed that the transient and steady-state deformation of capsules was significantly affected by inertia, and the flow structure varied considerably when Re was varied from 0.1 to 10. No significant changes were found for Re ranging from 0.01 to 0.1, and hence the inertia effect on capsule deformation in the microscale flows can be omitted when Re is less than 0.1. These findings improve the current understanding of the mechanism underlying cell movement in capillaries and can be applied to optimize the conditions for cell manipulation and separation in microfluidic devices.     

Computer simulation of heat,mass and fluid flows in a melt during czochralski crystal growth

A procedure to obtain steady state heat, mass and fluid flows in the melt during Czochralski crystal growth is described. This method is satisfactorily applied to the isothermal fluid flows caused by crystal rotation and to the nonisothermal fluid flows caused by combined free and forced convection. For high Reynolds number Re = a²ω_s/v, where a is the crystal radius, ω_s is the crystal rotation rate, and v is the kinematic viscosity, this method is revised so as to avoid numerical divergence.     

Multiple vortex formation in steady laminar axisymmetric impinging flow

Steady confined laminar axisymmetric impinging flow of a Newtonian fluid is relevant in many situations, an important application being heat and mass transfer from a solid surface to an impinging jet. This paper focuses on the evolution of the structure of the radial flow field in the channel region beyond the impingement zone. We employ an upwind scheme with an established numerical technique to solve the stream function and vorticity equations for a range of Reynolds numbers Re and geometrical aspect ratios e. Our results show the progressive complexity in the radial flow due to multiple points of flow separation and reattachment, and we provide a detailed demarcation of the Re-e plane based on flow separation behavior. In addition to the primary and secondary vortices anchored on the confining and impinging surfaces, respectively, we describe the formation and properties of a tertiary vortex which is wholly enclosed within the primary vortex. At a fixed Reynolds number, the tertiary vortex is observed only for a specific range of the aspect ratio, and we catalog its birth, growth and demise as the aspect ratio is varied. The range of aspect ratios over which the tertiary vortex exists is seen to increase with the Reynolds number. These results show that the fine structure of the radial flow at high Reynolds number continues to be dependent on the aspect ratio in a complex manner. At a given aspect ratio, the sizes of the vortices increases with Reynolds number, scaling as ∼Re^1/3, and for sufficiently large Re, the length of the tertiary vortex can exceed that of the secondary vortex. The primary and secondary vortex lengths satisfy an asymptotic relationship independent of Re and e, the numerically computed value of α being ∼2. Similarly, the locations of these vortices bear simple linear relationships independent of Re and e. Furthermore, despite the complex fine structure of the flow field, macroscopic flow properties such as vortex circulation and excess pressure loss continue to exhibit relatively simple dependence on Re and e, in accordance with previous results at much lower Reynolds numbers. Finally, some comments are made regarding the possibility of additional cascaded or isolated vortices occurring at even higher Reynolds numbers and aspect ratios.     

Finite element solution of viscous jet flows with surface tension

A finite element method is used to study the effect of Reynolds number and surface tension on the expansion and contraction of jets of Newtonian liquids. For values of Reynolds numbers (based on tube diameter), below 14 the jets expand, and when Re > 14 the jets contract. For higher Reynolds numbers the jet diameter approaches a limiting value. It is also found that the surface tension has a considerable effect on low Reynolds number jet flows, becoming negligible at higher Reynolds numbers. As an example, if the surface tension parameter $\text{σ}\text{η}\text{u}$ is equal to unity, the creeping flow jet expansion is reduced by 4% relative to the case with no surface tension but when Re is equal to 20 and 50 the final jet diameters increase by only 0.2%. The calculated jet shapes are compared with available experimental results.     

Wall modeling for LES of high Reynolds number channel flows: What turbulence information is retained?

A novel near-wall eddy-viscosity formulation for Large-eddy simulation (LES) has been used to compute high Reynolds number channel flows up to Re_τ = 1,000,000. These computations allow an insight into what turbulence information is retained when LES with a wall model is applied to such high Reynolds numbers. Detailed results are presented for the mean and rms velocities, as well as energy spectra. It is observed that, when an appropriate scaling is used, the rms velocities, energy spectra and the production of turbulence kinetic energy are weakly Reynolds number dependent at these high Reynolds numbers.     

Large-eddy simulation of streamwise-rotating turbulent channel flow

In many engineering and industrial applications the investigation of rotating turbulent flow is of great interest. Whereas some research has been done concerning channel flows with a spanwise rotation axis, only few investigations have been performed on channel flows with a rotation about the streamwise axis. In the present study an LES of a turbulent streamwise-rotating channel flow at Re_τ = 180 is performed using a moving grid method. The three-dimensional structures and the details of the secondary flow distribution are analyzed and compared with experimental data. The numerical-experimental comparison shows a convincing agreement as to the overall flow features. The results confirm the development of a secondary flow in the spanwise direction, which has been found to be correlated to the rotational speed. Furthermore, the findings show the distortion of the main flow velocity profile, the slight decrease of the streamwise Reynolds stresses in the vicinity of the walls, and the pronounced increase of the spanwise Reynolds stresses at higher rotation rates near the walls and particularly in the symmetry region. As to the numerical set-up it is shown that periodic boundary conditions in the spanwise direction suffice if the spanwise extent of the computational domain is larger than 10 times the channel half width.     

Spectral Vanishing Viscosity Method for Large-Eddy Simulation of Turbulent Flows

An efficient spectral vanishing viscosity method for the large-eddy simulation of incompressible flows is proposed, both for standard spectral and spectral element approximations. The approach is integrated in a collocation spectral Chebyshev-Fourier solver and then used to compute the turbulent wake of a cylinder in a crossflow confined geometry (Reynolds number Re=3900)