Time-grid independent error analysis of adaptive predictor-corrector BDF2 scheme for the unsteady Navier–Stokes equations with high Reynolds number

Imai suggested a new method to estimate the pressure drag coefficients of bluff bodies at high Reynolds number. He supposed that the flow at high Reynolds number might be obtained by solving a modified Navier-Stokes equation at low Reynolds number which was determined by the assumption of the eddy viscosity in the wake region. In this paper authors discuss the details of the two-dimensional viscous fluid flows past the blunt bodies of arbitrary shape at low Reynolds number (R=40) by solving numerically Navier-Stokes equations, and investigate the validity of Imai's hypothesis above mentioned.The results obtained are that Imai's idea concerning the flow pattern can be acceptable, and that the pressure drag coefficients of bluff bodies at low Reynolds number agree approximately with those in the high but subcritical Reynolds number range. But that Imai's idea gives invalid informations as to the pressure distribution along the body surface at high Reynolds number.

In this paper, we propose and study first- and second-order (in time) stabilized linear finite element schemes for the incompressible Navier-Stokes (NS) equations. The energy, momentum, and angular momentum conserving (EMAC) formulation has emerged as a promising approach for conserving energy, momentum, and angular momentum of the NS equations, while the exponential scalar auxiliary variable (ESAV) has become a popular technique for designing linear energy-stable numerical schemes. Our method leverages the EMAC formulation and the Taylor-Hood element with grad-div stabilization for spatial discretization. We adopt the implicit-explicit backward differential formulas (BDFs) coupled with a novel stabilized ESAV approach for time stepping. For the solution process, we develop an efficient decoupling technique for the resulting fully-discrete systems so that only one linear Stokes solve is needed at each time step, which is similar to the cost of classic implicit-explicit BDF schemes for the NS equations. Robust optimal error estimates are successfully derived for both velocity and pressure for the two proposed schemes, with Gronwall constants that are particularly independent of the viscosity. Furthermore, it is rigorously shown that the grad-div stabilization term can greatly alleviate the viscosity-dependence of the mesh size constraint, which is required for error estimation when such a term is not present in the schemes. Various numerical experiments are conducted to verify the theoretical results and demonstrate the effectiveness and efficiency of the grad-div and ESAV stabilization strategies and their combination in the proposed numerical schemes, especially for problems with high Reynolds numbers.

The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

Adaptive finite element simulations of the surface currents in the North Sea

Acceleration of unsteady hydrodynamic simulations using the parareal algorithm

Fluid flow, heat transfer and entropy generation characteristics of micro-pipes are investigated computationally by considering the simultaneous effects of pipe diameter, wall heat flux and Reynolds number in detail. Variable fluid property continuity, Navier-Stokes and energy equations are numerically handled for wide ranges of pipe diameter (d = 0.50–1.00 mm), wall heat flux (q''= 1000–2000 W/m2) and Reynolds number (Re = 1 – 2000), where the relative roughness is kept constant at e/d = 0.001 in the complete set of the scenarios considered. Computations indicated slight shifts in velocity profiles from the laminar character at Re = 500 with the corresponding shape factor (H) and intermittency values (γ) of H = 3.293→3.275 and γ = 0.041→0.051 (d = 1.00→0.50 mm). Moreover, the onset of transition was determined to move down to Retra = 1,656, 1,607, 1,491, 1,341 and 1,272 at d = 1.00, 0.90, 0.75, 0.60 and 0.50 mm, respectively. The impacts of pipe diameter on friction mechanism and heat transfer rates are evaluated to become more significant at high Reynolds numbers, resulting in the rise of energy loss data at the identical conditions as well. In cases with low pipe diameter and high Reynolds number, wall heat flux is determined to promote the magnitude of local thermal entropy generation rates. Local Bejan numbers are inspected to rise with wall heat flux at high Reynolds numbers, indicating that the elevating role of wall heat flux on local thermal entropy generation is dominant to the suppressing function of Reynolds number on local thermal entropy generation. Cross-sectional total entropy generation is computed to be most influenced by pipe diameter at high wall heat flux and low Reynolds numbers.

Heat transfer in a rotating lateral outflow trapezoidal channel with pin fins under high rotation numbers and Reynolds numbers

An experiment has been performed to study the aerodynamics of dynamic stall penetration at constant pitch rate and high Reynolds number, in an attempt to model more accurately conditions during aircraft poststall maneuvers and during helicopter high-speed forward flight. An airfoil was oscillated at pitch rates, A = ac/2U between 0.001 and 0.020, Mach numbers between 0.2 and 0.4, and Reynolds numbers between 2-4 x 10. Surface pressures were measured using 72 miniature transducers, and the locations of transition and separation were determined using 8 surface hot-film gages. The results demonstrate the influence of the leading-edge vorticity on the unsteady aerodynamic response during and after stall. The vortex is strengthened by increasing the pitch rate and is weakened by increasing the Mach number and by starting the motion close to the steady-state stall angle. A periodic pressure oscillation occurred after stall at high pitch angle and moderate Reynolds number; the oscillation frequency was close to that predicted for a von Karman vortex street. A small supersonic zone near the leading edge at M = 0.4 was found to reduce significantly the peak suction pressures and the unsteady increments to the airloads. These results provide the first known data base of constant-pitch-rate aerodynamic information at realistic combinations of Reynolds and Mach numbers.

Effect of tip clearance variation in the transonic axial compressor of a miniature gas turbine at different Reynolds numbers

Laboratory experiments are performed to understand the controlling parameters of the electrical field associated with the seepage of water through a porous material. We use seven glass bead packs with varying mean grain size in an effort to obtain a standard material for the investigation of these electrical potentials. The mean grain size of these samples is in the range 56–3000 μm. We use pure NaCl electrolytes with conductivity in the range 10−4 to 10−1 S m−1 at 25°C. The flow conditions cover viscous and inertial laminar flow conditions but not turbulent flow. In the relationship between the streaming potential coupling coefficient and the grain size, three distinct domains are defined by the values of two dimensionless numbers, the Dukhin and the Reynolds numbers. The Dukhin number represents the ratio between the surface conductivity of the grains (due to conduction in the electrical double layer coating the surface of the grains) and the pore water electrical conductivity. At high Dukhin numbers (≫1) and low Reynolds numbers (≪1), the magnitude of the streaming potential coupling coefficient decreases with the increase of the Dukhin number and depends on the mean grain diameter (and therefore permeability) of the medium. At low Dukhin and Reynolds numbers (≪1), the streaming potential coupling coefficient becomes independent of the microstructure and is given by the well‐known Helmholtz‐Smoluchowski equation widely used in the literature. At high Reynolds numbers, the magnitude of the streaming potential coupling coefficient decreases with the increase of the Reynolds number in agreement with a new model developed in this paper. A numerical application is made illustrating the relation between the self‐potential signal and the intensity of seepage through a leakage in an embankment.

Local and global effects of cylindrical vortex generators on the mass transfer distributions over the four active walls of a square, rib-roughened rotating duct with a sharp 180° bend are investigated. Cylindrical vortex generators (rods) are placed above, and parallel to, every other rib on the leading and trailing walls of the duct so that their wake can interact with the shear layer and recirculation region formed behind the ribs, as well as the rotation-generated secondary flows. Local increases in near-wall turbulence intensity resulting from these interactions give rise to local enhancement of mass (heat) transfer. Measurements are presented for duct Reynolds numbers (Re) in the range 5000–30,000, and for rotation numbers in the range 0 to 0.3. The rib height-to-hydraulic diameter ratio (e/Dh) is fixed at 0.1, while the rib pitch-to-rib height ratio (P/e) is 10.5. The vortex generator rods have a diameter-to-rib height ratio (d/e) of 0.78, and the distance separating them from the ribs relative to the rib height (s/e) is 0.55. Mass transfer measurements of naphthalene sublimation have been carried out using an automated acquisition system and are correlated with heat transfer using the heat/mass transfer analogy. The results indicate that the vortex generators tend to enhance overall mass transfer in the duct, compared to the case where only ribs are present, both before and after the bend at high Reynolds and Rotation numbers. Local enhancements of up to 30% are observed on all four walls of the duct. At low Reynolds numbers (e.g. 5,000) the insertion of the rods often leads to degradation. At high Reynolds numbers (e.g. 30,000) the enhancement due to the rods occurs on the surfaces stabilized by rotation (trailing edge on the inlet pass and leading edge on the outlet pass) and the side walls.. The enhancement is more pronounced as the Rotation number is increased. The detailed measurements in a ribbed duct with vortex-generator rods clearly show localized regions of enhanced mass (heat) transfer at Reynolds and Rotation numbers within the envelope of practical interest for gas-turbine blade cooling applications.

Updates on the Implicit Solver for Incompressible Navier-StokesAccurate numerical simulations of unsteady fluid flows require high-fidelity methods such as large eddy simulation and direct numerical simulation. Both methods are, however, limited in practical applications due to high computational costs. While the straightforward way to escape this issue is using more computational resources, cost and time constraints often require algorithmic improvements instead. High-fidelity methods typically rely on explicit time-stepping schemes due to low computational cost per time step. In this work, we investigate an implicit time-stepping scheme that increases the computational cost for each time step, but permits larger time steps to reduce the total time-to-solution. In particular, we study a class of fractional step schemes known as velocity correction schemes that are commonly employed for solving the incompressible Navier-Stokes equations. The workhorse semi-implicit velocity correction scheme uses implicit diffusion and explicit advection treatment. The explicit advection treatment becomes a performance bottleneck for high Reynolds number flows around complex geometries, because of a CFL-type condition that constrains the algorithm’s stability. We explore a linear-implicit velocity correction scheme which removes the CFL limitation. The linear-implicit scheme uses a linearisation of the advection operator and, hence, preserves the linear structure of the semi-implicit algorithm. As such the approach efficiently improves stability without significant computational overhead. We perform a comparison of both, the semi-implicit and linear-implicit, velocity correction schemes and look into their stability, accuracy and computational performance. Our investigation includes canonical problems and a low Reynolds number and two-dimensional cylin- der flow for verification. Further, we investigate turbulent flows at high Reynolds number and around challenging geometries based on Formula 1 race cars. We find that the linear-implicit scheme achieves strong improvements in the stability allowing up to 100-times larger time step sizes on the most complex 3D geometry. The influence on the accuracy with larger time step sizes varies and all cases show negligible differences up to at least 10-times larger time step sizes. Additionally, the stability improvement provides leverage for the computational performance enabling a 5-fold speed-up without loss in accuracy and a 10-fold speed-up in time-to-solution with minor losses in the accuracy. A Level Set Multiphase Solver framework in Nektar++Level set equations coupled with Navier-Stokes equations are implemented into the Nektar++ framework to model the interface dynamics and surface tension in order to produce a spectral/hp multiphase solver. This multiphase solver is built on top of the existing Incompressible Navier-Stokes Solver (IncNavierStokesSolver) and handles two fluids (liquid or gas) with different physical properties. The level set field is treated as a smeared out Heaviside function with values in the range of 0 to 1 representing the diffuse interface. Following the explicit advection of the level set function, the reinitialisation of the level set function is achieved through the solution of Helmholtz form of the remaining level set equation with addition of artificial compressive flux and a small viscosity value proportional to the grid resolution. The diffused interface level set field is utilised to model the interface curvature and surface tension as a continum surface force (CSF) accounted for in the momentum conservation. As a first step, the results of the benchmark test cases of advection (rotating circle) and deformation (vortex) as well as Zalesak's slotted disk and 3D sphere deformation from the Continuous Galerkin form of level set function solution, revealed that the spectral/hp framework Nektar++ was able to produce accurate results and limited numerical diffusion even in coarse meshes. The performance of the current version of the solver is presented with sample cases of liquid droplet deformation under uniform gas flow and the effect of surface tension.

Experimental investigations on liquid metal MHD turbulent flows through a circular pipe with a conductive wall

Physical experiments are used to explore the properties of the motions contributing to the Reynolds stresses in high and low Reynolds number turbulent boundary layers. The low Reynolds number smooth wall measurements (Rθ=1010, Rθ=2870, and Rθ=4850) were acquired in a large-scale low speed wind tunnel, while the high Reynolds number measurements [Rθ∼O(106)] were acquired at the Surface Layer Turbulence and Environmental Science Test site, Dugway, Utah. These high Reynolds number turbulent boundary layer data were acquired over nearly hydraulically smooth and rough walls. At each Reynolds number and surface roughness, data comparisons were made at approximately yp/2 and 2yp, where yp is the peak position of the Reynolds shear stress. Scale separation effects associated with increasing Rθ are analyzed via spectral measurements (u, v, and u-v cospectra), and by segregating the streamwise and wall-normal velocities according to their frequency content using simultaneous high- and low-pass filtering. A primary observation is that the predominant motions underlying the stress undergo a significant shift from large to intermediate scales as Rθ becomes large, irrespective of surface roughness. Quadrant analysis of the filtered signals is employed to clarify the correlated scales involved in the generation of the stress. Overall, it is apparent that the types of motions contributing to Reynolds stress undergo significant variations at comparable wall-normal locations (relative to yp) over the Reynolds number range explored.

In this paper, we analyze the first-order implicit-explicit type scheme based on the scalar auxiliary variable (SAV) with divergence-free H 1 H^1 conforming finite element method (FEM) in space for the evolutionary incompressible Navier-Stokes equations at high Reynolds number. The stability and a priori error estimates are given, in which the constants are independent of the Reynolds number. The velocity energy estimate is given without any condition on the time step, however, the a priori error estimates for the velocity are obtained with severe time step restrictions. In addition, a Reynolds-dependent error bound with convergence order of k + 1 k+1 in space is also obtained for the velocity error in the L 2 L^2 norm with no time step restrictions. Here, k k is the polynomial order of the velocity space. Some numerical experiments are carried out to verify the analytical results.