Effects of Reynolds number on physiological‐type pulsatile flows in a pipe with ring‐type constrictions

The effects of Reynolds number on the physiological‐type of laminar pulsatile flow fields within the vicinity of mechanical ring‐type constriction in small pipes were studied numerically. The parameters considered are: the Reynolds number (Re) in the range of 50–1500; Strouhal number (St) in the range of 0.00156–3.98; Womersley number (Nw) from 0.0 to 50.0. The pulsatile flows considered were physiological‐type of simulated flows. Within a pulsating cycle, detailed flow characteristics were studied through the pulsating contours of streamline (ψ), vorticity (Ω), shear stress (τ) and isobar. The relations between the instantaneous flow rate (Q) and instantaneous pressure gradients (dp/dz) are observed to be elliptic. The relations between the instantaneous flow rate (Q) and pressure loss (P_loss) are quadratic. Linear relations were observed between the instantaneous flow rate (Q) and the maximum velocity, maximum vorticity and maximum shear stress. The Reynolds number of the flow in a pulsating cycle was found to have significant effects on the recirculation length and the pressure gradient within the pulsatile flow regime. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical study of turbulent flow over two‐dimensional surface‐mounted ribs in a channel

This study accurately predicts the cases of turbulent flow around a surface‐mounted two‐dimensional rib with varying lengths. The numerical method employs a differencing scheme for integrating the elliptic Reynolds‐averaged Navier–Stokes equations and the continuity equation. A two‐equation k–ε turbulence model is employed to simulate the turbulent transport quantities and close the solving problem. The near‐wall regions of the separated sides of the rib are resolved by a near‐wall model of a two‐layer approach instead of the wall function approximation. Computations for flow over a surface‐mounted rectangular rib are conducted for the variations in the rib lengths. Results indicate that upstream of the obstacle, the length of the recirculating region remains unchanged with varying rib lengths; while the downstream length of the recirculating region is a strong function of rib length and changes nearly linearly for the varying lengths of B/H=0.1 to B/H=4.0. Reattachment on top of the rib, owing to its increasing length, affects the downstream boundary layer development. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical study of pulsatile laminar flows in a pipe with a ring-type constriction

Numerical simulations have been carried out to study pulsatile laminar flows in a pipe with an axisymmetric ringtype constriction. Three types of pulsatile flows were investigated, namely a physiological flow, a pure sinusoidal flow and a non-zero mean velocity sinusoidal flow. The laminar flow governing equations were solved by the SIMPLE algorithm on a non-staggered grid and a modified Crank-Nicolson approximation was used to discretrize the momentum equations with respect to time. The maximum flow Reynolds numer (Re) is 100. The Womersley number (N_w) ranges from 0 to 50, with the corresponding Strouhal number (St) ranging from 0 to 3·98. The constriction opening ratio (d/D) and thickness ratio (h/D) are fixed at 0·5 and 0·1 respectively. Within the time period investigated, all these pulsatile flows include both forward and backward flows. The unsteady recirculation region and the recirculation points change in size and location with time. For N_w ≤ 1 and St≤ 1·56 x 10^?3 the three pulsatile flows have the same simple relation between the instantaneous flow rate and pressure loss (Δp) across the constriction and the pressure gradient in the axial direction (dp/dz) in the fully developed flow region. The phase angles between the flow rate and pressure loss and the pressure gradient are equal to zero. With increasing N_w and St, the phase angle between the flow rate and the dp/dz becomes larger and has its maximum value of 90° at N_w = 50 and St = 3·98. The three pulsatile flows also show different relations between the flow rate and the pressure gradient. The pure sinusoidal flow has the largest maximum pressure gradient and the non-zero mean velocity sinusoidal flow has the smallest. For larger N_w and St the fully developed velocity profiles in the fully developed flow region have a smaller velocity gradient along the radial direction in the central region. The maximum recirculation length increases for N_w ranging from 0 to 4·2, while this length becomes very small at N_w = 50 and St = 3·98. The deceleration tends to enlarge the recirculation region and this effect appears for N_w ≥ 3 and St ≥ 1·43×10^?2. Linear relations exist between the flow rate and the instantaneous maximum values of velocity, vorticity and shear stress.     

A numerical study of turbulent flow over a two‐dimensional hill

Calculations of mean velocities and Reynolds stresses are reported for the recirculating flow established in the wake of two‐dimensional polynomial‐shaped obstacles that are symmetrical about a vertical axis and mounted in the water channel downstream of a fully developed channel flow for Re=6×10⁴. The study involves calculations of mean and fluctuating flow properties in the streamwise and spanwise directions and include comparisons with experimental data [Almeida GP, Durão DFG, Heitor MV. Wake flows behind two‐dimensional model hills. Experimental Thermal and Fluid Science 1993; 7: 87–101] for flow around a single obstacle with data resulting from the interaction of consecutive obstacles, using two versions of the low‐Reynolds number differential second‐moment (DSM) closure model. The results include analysis of the turbulent stresses in local flow co‐ordinates and reveal flow structure qualitatively similar to that found in other turbulent flows with a reattachment zone. It is found that the standard isotropization of production model (IPM), based on that proposed by Gibson and Launder [Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 1978; 86(3): 191–511], with the incorporation of the wall reflection model of Craft and Launder [New wall‐reflection model applied to the turbulent impinging jet. AIAA Journal 1992; 32(12): 2970–2972] predicts the mean velocities quite well, but underestimates the size of the recirculation region and turbulent quantities in the shear layer. These inadequacies are circumvented by adopting a new cubic Reynolds stress closure scheme based on that more recently developed by Craft and Launder [A Reynolds stress closure designed for complex geometries. International Journal of Heat and Fluid Flow 1996; 17: 245–254] which satisfies the two component limit (TCL) of turbulence. In this model the geometry‐specific quantities, such as the wall‐normal vector or wall distance, are replaced by invariant dimensionless gradient indicators. Also, the model captures the diverse behaviour of the different components of the stress dissipation, ε_ij, near the wall and uses a novel decomposition for the fluctuating pressure terms. Copyright © 2001 John Wiley & Sons, Ltd.     

Residual‐based variational multiscale methods for laminar,transitional and turbulent variable‐density flow at low Mach number

In the present study, residual‐based variational multiscale methods are developed for and applied to variable‐density flow at low Mach number. In particular, two different formulations are considered in this study: a standard stabilized formulation featuring SUPG/PSG/grad‐div terms and a complete residual‐based variational multiscale formulation additionally containing cross‐ and Reynolds‐stress terms as well as subgrid‐scale velocity terms in the energy‐conservation equation. The proposed methods are tested for various laminar flow test cases as well as a test case at laminar via transitional to turbulent flow stages. Stable and accurate results are obtained for all numerical examples. Substantial differences in the results between the two approaches do not become notable until a high temperature gradient is applied and the flow reaches a turbulent flow stage. The more pronounced influence of adding subgrid‐scale velocity terms to the energy‐conservation equation on the results than adding analogous terms to the momentum‐conservation equation in this situation appears particularly noteworthy. Copyright © 2009 John Wiley & Sons, Ltd.     

An algebraic variational multiscale–multigrid method for large‐eddy simulation of turbulent pulsatile flows in complex geometries with detailed insight into pulmonary airway flow

The algebraic variational multiscale–multigrid method, an advanced computational approach recently proposed for large‐eddy simulation of turbulent flow, is further developed in this study for turbulent flow simulations in complex geometries. In particular, it is applied to the complex case of pulsatile turbulent flow dynamics of the upper and lower pulmonary airways up to generation 7 and carefully investigated for this important application. Among other things, the results obtained with the proposed method are compared with the results obtained with a rather traditional stabilized finite element method. As opposed to previous large‐eddy simulations of pulmonary airways, we consider a pulsatile inflow condition, allowing the development of turbulence over a pulse cycle to be investigated, which obviously makes these results more physiologically realistic. Our results suggest that turbulent effects in the bronchial airways are rather weak and can completely decay as early as the third generation, depending on geometry and flow distribution. Both methods utilized in this study are able to adequately capture all flow stages from laminar via transitional to turbulent regimes without any modifications. However, the algebraic variational multiscale–multigrid method provides superior results as soon as the flow enters the most challenging, turbulent flow regime. Furthermore, the robustness of the scale‐separation approach based on plain aggregation algebraic multigrid inherent to the algebraic variational multiscale–multigrid method is demonstrated for the present complex geometry. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical prediction of turbulent flow structure generated by a synthetic cross‐jet into a turbulent boundary layer

A detailed numerical study using large‐eddy simulation (LES) and unsteady Reynolds‐averaged Navier–Stokes (URANS) was undertaken to investigate physical processes that are engendered in the injection of a circular synthetic (zero‐net mass flux) jet in a zero pressure gradient turbulent boundary layer. A complementary study was carried out and was verified by comparisons with the available experimental data that were obtained at corresponding conditions with the aim of achieving an improved understanding of fluid dynamics of the studied processes. The computations were conducted by OpenFOAM C++, and the physical realism of the incoming turbulent boundary layer was secured by employing random field generation algorithm. The cavity was computed with a sinusoidal transpiration boundary condition on its floor. The results from URANS computation and LES were compared and described qualitatively and quantitatively. There is a particular interest for acquiring the turbulent structures from the present numerical data. The numerical methods can capture vortical structures including a hairpin (primary) vortex and secondary structures. However, the present computations confirmed that URANS and LES are capable of predicting current flow field with a more detailed structure presented by LES data as expected. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of compressible turbulent flow via improved gas‐kinetic BGK scheme

In order to solve compressible turbulent flow problems, this study focuses on incorporating the Spalart–Allmaras turbulence model into gas‐kinetic BGK (Bhatnagar–Gross–Krook) scheme. The Spalart–Allmaras turbulence model is solved using finite difference discretization. The variables on the cell interface are interpolated via the van Leer limiter in the reconstruction stage. Simulation of subsonic and transonic flow over a NACA0012 airfoil has been implemented using two‐dimensional body‐fitted grids. The numerical results obtained appear in good agreement with the AGARD results, demonstrating the effectiveness and usefulness of the strategy of coupling the Spalart–Allmaras turbulence model with the BGK scheme for compressible turbulent flow simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical computation of turbulent flow through a square-sectioned 90° bend

Finite-volume, semi-elliptic computations are reported of the three-dimensional flow around a 90° square-sectioned bend for which detailed laser-Doppler measurements have been reported by Taylor et al.¹ While the standard k- eddy-viscosity model has been used in the main flow region, in place of the usual “wall functiond”, the mixing-length hypothesis has been employed to resolve the flow in the layer immediately adjacent to the wall. The scheme is successful in predicting the details of the primary and secondary flow fields both within the bend and downstream thereof.     

Numerical analysis of turbulent flow separation in a rectangular duct with a sharp 180‐degree turn by algebraic Reynolds stress model

Turbulent flow in a rectangular duct with a sharp 180‐degree turn is difficult to predict numerically because the flow behavior is influenced by several types of forces, including centrifugal force, pressure‐driven force, and shear stress generated by anisotropic turbulence. In particular, this type of flow is characterized by a large‐scale separated flow, and it is difficult to predict the reattachment point of a separated flow. Numerical analysis has been performed for a turbulent flow in a rectangular duct with a sharp 180‐degree turn using the algebraic Reynolds stress model. A boundary‐fitted coordinate system is introduced as a method for coordinate transformation to set the boundary conditions next to complicated shapes. The calculated results are compared with the experimental data, as measured by a laser‐Doppler anemometer, in order to examine the validity of the proposed numerical method and turbulent model. In addition, the possibility of improving the wall function method in the separated flow region is examined by replacing the log‐law velocity profile for a smooth wall with that for a rough wall. The analysis results indicated that the proposed algebraic Reynolds stress model can be used to reasonably predict the turbulent flow in a rectangular duct with a sharp 180‐degree turn. In particular, the calculated reattachment point of a separated flow, which is difficult to predict in a turbulent flow, agrees well with the experimental results. In addition, the calculation results suggest that the wall function method using the log‐law velocity profile for a rough wall over a separated flow region has some potential for improving the prediction accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

Convergence acceleration by self‐adjusted time stepsize using Bi‐CGSTAB method for turbulent separated flow computation

Poor convergence behavior is usually encountered when numerical computations on turbulent separated flow are performed. A design of self‐adjusted stepsize concept both in time span and spatial coordinate systems to achieve faster convergence is demonstrated in this study. The determination of the time stepsize based on the concept of minimization of residuals using the Bi‐CGSTAB algorithm is proposed. The numerical results show that the time stepsize adjusted by the proposed method indeed improves the convergence rate for turbulent separated flow computations using advanced turbulence models in low‐Reynolds number forms. Copyright © 2002 John Wiley & Sons, Ltd.     

Surface‐sampled simulations of turbulent flow at high Reynolds number

A new approach to turbulence simulation, based on a combination of large eddy simulation (LES) for the whole flow and an array of non–space‐filling quasi‐direct numerical simulations (QDNS), which sample the response of near‐wall turbulence to large‐scale forcing, is proposed and evaluated. The technique overcomes some of the cost limitations of turbulence simulation, since the main flow is treated with a coarse‐grid LES, with the equivalent of wall functions supplied by the near‐wall sampled QDNS. Two cases are tested, at friction Reynolds number Re_τ=4200 and 20000. The total grid point count for the first case is less than half a million and less than 2 million for the second case, with the calculations only requiring a desktop computer. A good agreement with published direct numerical simulation (DNS) is found at Re_τ=4200, both in the mean velocity profile and the streamwise velocity fluctuation statistics, which correctly show a substantial increase in near‐wall turbulence levels due to a modulation of near‐wall streaks by large‐scale structures. The trend continues at Re_τ=20000, in agreement with experiment, which represents one of the major achievements of the new approach. A number of detailed aspects of the model, including numerical resolution, LES‐QDNS coupling strategy and subgrid model are explored. A low level of grid sensitivity is demonstrated for both the QDNS and LES aspects. Since the method does not assume a law of the wall, it can in principle be applied to flows that are out of equilibrium.     

Numerical study of the turbulent flow in strongly curved stationary and rotating U‐ducts

A large‐eddy simulation (LES) study is undertaken to explore the complex flow of developing turbulent flow through stationary and rotating U‐ducts with strong curvature. Three flow cases are investigated: stationary (non‐rotating), positive and negative rotational cases. Stationary and positive rotational cases are shown to have similar flow characteristics in terms of the mean velocity variations, although the predicted separation zone is nearly doubled in size for positive rotational case. Unlike the positive rotation for which the laminarization effects are observed, turbulence is significantly enhanced for the negative rotation mainly due to the existence of strong secondary flow. Turbulence is found to be highly anisotropic throughout the duct apart from the far downstream regions of the bend for the negative rotational case. The stress–strain relation seems to be completely invalid in the U‐duct apart from the bend region. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical and experimental investigation of turbulent flow in a rectangular duct

Details of the turbulent flow in a 1:8 aspect ratio rectangular duct at a Reynolds number of approximately 5800 were investigated both numerically and experimentally. The three-dimensional mean velocity field and the normal stresses were measured at a position 50 hydraulic diameters downstream from the inlet using laser doppler velocimetry (LDV). Numerical simulations were carried out for the same flow case assuming fully developed conditions by imposing cyclic boundary conditions in the main flow direction. The numerical approach was based on the finite volume technique with a non-staggered grid arrangement and the SIMPLEC algorithm. Results have been obtained with a linear and a non-linear (Speziale) k–ε model, combined with the Lam–Bremhorst damping functions for low Reynolds numbers. The secondary flow patterns, as well as the magnitude of the main flow and overall parameters predicted by the non-linear k–ε model, show good agreement with the experimental results. However, the simulations provide less anisotropy in the normal stresses than the measurements. Also, the magnitudes of the secondary velocities close to the duct corners are underestimated. © 1998 John Wiley & Sons, Ltd.     

Numerical study of an oscillatory turbulent flow over a flat plate

Oscillatory turbulent flow over a flat plate is studied using large eddy simulation (LES) and Reynolds-average Navier-Stokes (RANS) methods. A dynamic subgrid-scale model is employed in LES and Saffman's turbulence model is used in RANS. The flow behaviors are discussed for the accelerating and decelerating phases during the oscillating cycle. The friction force on the wall and its phase shift from laminar to turbulent regime are also investigated for different Reynolds numbers. The project supported by the Youngster Funding of Academia Sinica and by the National Natural Science Foundation of China     

Energy dynamics in a turbulent channel flow using the Karhunen‐Loéve approach

The dynamical equations for the energy in a turbulent channel flow have been developed by using the Karhunen‐Loéve modes to represent the velocity field. The energy balance equations show that all the energy in the flow originates from the applied pressure gradient acting on the mean flow. Energy redistribution occurs through triad interactions, which is basic to understanding the dynamics. Each triad interaction determines the rate of energy transport between source and sink modes via a catalyst mode. The importance of the proposed method stems from the fact that it can be used to determine both the rate of energy transport between modes as well as the direction of energy flow. The effectiveness of the method in determining the mechanisms by which the turbulence sustains itself is demonstrated by performing a detailed analysis of triad interactions occurring during a turbulent burst in a minimal channel flow. The impact on flow modification is discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

Acoustic upwinding for sub‐ and super‐sonic turbulent channel flow at low Reynolds number

A recently developed asymmetric implicit fifth‐order scheme with acoustic upwinding for the spatial discretization for the characteristic waves is applied to the fully compressible, viscous and non‐stationary Navier–Stokes equations for sub‐ and super‐sonic, mildly turbulent, channel flow (Re_τ=360). For a Mach number of 0.1, results are presented for uniform (32³, 64³ and 128³) and non‐uniform (expanding wall‐normal, 32³ and 64³) grids and compared to the (incompressible) reference solution found in (J. Fluid. Mech. 1987; 177 :133–166). The results for uniform grids on 128³ and 64³ nodes show high resemblance with the reference solution. Expanding grids are applied on 64³‐ and 32³‐node grids. The capability of the proposed technique to solve compressible flow is first demonstrated by increasing the Mach number to 0.3, 0.6 and 0.9 for isentropic flow on the uniform 64³‐grid. Next, the flow speed is increased to Ma=2. The results for the isothermal‐wall supersonic flows give very good agreement with known literature results. The velocity field, the temperature and their fluctuations are well resolved. This means that in all presented (sub‐ and super‐sonic) cases, the combination of acoustic upwinding and the asymmetric high‐order scheme provides sufficient high wave‐number damping and low wave‐number accuracy to give numerically stable and accurate results. Copyright © 2007 John Wiley & Sons, Ltd.