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Direct numerical simulation of internal compressible flows at high Reynolds number: numerical and physical insight
Full text linkThis work reports results of direct numerical simulations (DNS) of compressible internal flows. For this purpose three internal flow geometries of increasing complexity are considered, namely planar channel, pipe and rectangular duct flow. The work focuses on both numerical and physical issues related to wall-bounded turbulent flows. In the first part of the work some numerical issues concerning the solution of compressible wall-bounded flows, both in Cartesian and cylindrical coordinates, are addressed. Attention is focused on the acoustic time-step limitation which, in the case of wall-bounded flows, is restrictive across all Mach numbers. For this reason we develop a semi-implicit algorithm for time-accurate simulation of the compressible Navier-Stokes (N-S) equations. The method is based on linearization of the partial convective fluxes associated with acoustic waves, in such a way to suppress, or at least mitigate the acoustic time step restriction. Together with replacement of the total energy equation with the entropy transport equation, this approach avoids the inversion of block-banded matrices involved in classical methods, which is replaced by less demanding inversion of standard banded matrices. This novel implementation, in which only Acoustic Terms are Implicit (ATI), is more efficient than previous approaches, barely requiring the inversion of a banded scalar system in each coordinate direction. All available data support higher computational efficiency than existing methods, and saving of resources ranging from 85% under low-subsonic flow conditions, to about 50% in supersonic flow. Numerical issues arising from the use of cylindrical coordinates are also discussed. We show that N-S equations in cylindrical coordinates can be conveniently recast to guarantee discrete conservation of total kinetic energy. The ATI approach is extended to the cylindrical case to deal with the severe time-step limitation in the azimuthal direction. In the second part of the work attention is focused on the effects of Mach and Reynolds number variation for the three flow geometries considered. DNS of planar channel, pipe and rectangular duct flow at bulk Mach number Mb=0.2,1.5, 3 and up to Re_tau=1000$ are presented. A long-standing topic in compressible flows is the relevant Reynolds number for comparing flow cases across the Mach number range. At this purpose, different compressibility transformations are compared to incompressible datasets at matching relevant Reynolds number. All data show that the Trettel-Larsson transformation allows excellent collapse of the compressible statistics on the incompressible ones, thus supporting the validity of semi-local scaling and Morkovin hypothesis. The size of the typical turbulent eddies is studied through spanwise spectral densities of the velocity field, which support validity of a scaling based on the local mean shear and the local friction velocity, with the main conclusion that the actual size of the eddies does not vary with the Mach number, at a fixed outer wall distance. Passive scalar transport is also studied across Mach and Reynolds number. Eventually, similarities and differences between compressible channel, pipe and rectangular duct flow are investigated
Reynolds and mach number effects in compressible turbulent channel flow
Full text linkThe effect of Reynolds and Mach number variation in compressible isothermal channel flow is investi-
gated through a series of direct numerical simulations (DNS), at bulk Mach number M b = 1.5, 3 and bulk
Reynolds number up to Re b = 34000, which is sufficient to sense sizeable high-Reynolds-number effects
not reached before in this type of flow. Dedicated incompressible DNS are also performed at precisely
matching Reynolds number, to directly gauge the performance of compressibility transformations for the
mean velocity profiles and Reynolds stresses. As in previous studies, we find inaccuracy of the classical
van Driest transformation to remove effects of variable density and viscosity, especially at low Reynolds
number. On the other hand, almost perfect matching of incompressible mean velocity and Reynolds stress
distributions is recovered throughout the wall layer by using a recently introduced transformation (Trettel
and Larsson, 2014,2016), the only remaining effect of compressibility being the increase of the streamwise
turbulence intensity peak with the Mach number. Temperature/velocity relations are scrutinized, with the
main finding that a recent relation by Zhang et al. (2014), which explicitly accounts for finite wall heat
flux, is more accurate than the classical Walz relation. The size of the typical turbulent eddies is studied
through spanwise spectral densities of the velocity field, which support validity of a scaling based on
the local mean shear and the local friction velocity, with the main conclusion that the actual size of the
eddies does not vary with the Mach number, at a fixed outer wall distance
Direct numerical simulation of forced thermal convection in square ducts up to
Full text linkWe carry out direct numerical simulation (DNS) of flow in a turbulent square
duct by focusing on heat transfer effects, considering the case of unit Prandtl
number. Reynolds numbers up to are considered which are
much higher than in previous studies, and which yield clear scale separation
between inner- and outer-layer dynamics. Close similarity between the behavior
of the temperature and the streamwise velocity fields is confirmed as in
previous studies related to plane channels and pipes. Just like the mean
velocity, the mean temperature is found to exhibit logarithmic layers as a
function of the nearest wall, however with a different slope. The most
important practical implication is the validity of the traditional hydraulic
diameter as the correct reference length for reporting heat transfer data, as
we rigorously show here. Temperature and velocity fluctuations also have
similar behavior, but apparently logarithmic growth of their inner-scaled peak
variances is not observed here unlike in canonical wall-bounded flows. Analysis
of the split contributions to the heat transfer coefficient shows that mean
cross-stream convection associated with secondary motions is responsible for
about of the total. Finally, we use the DNS database to highlight
shortcomings of traditional linear closures for the turbulent heat flux, and
show that substantial modeling improvement may be in principle obtained by
retaining at least the three terms in the vector polynomial integrity basis
expansion
On the suitability of the immersed boundary method for the simulation of high-Reynolds-number separated turbulent flows
No full textDelayed detached eddy simulation based on the Spalart–Allmaras turbulence model is applied in conjunction with the immersed boundary method to high-Reynolds number turbulent flows in complex geometries. A fourth-order, finite-difference solver capable of discretely preserving the total kinetic energy in the limit of inviscid flow is adopted to solve the compressible Navier–Stokes equations and model-consistent, adaptive wall functions are employed to provide proper numerical boundary conditions at the fluid/solid interface. Numerical tests, performed for several configurations involving massively separated flows, demonstrate that computations at high Reynolds number typically occurring in flows of industrial relevance can be successfully carried out using the immersed boundary strategy, providing predictions whose accuracy is comparable to that of standard, body-fitted, structured and unstructured flow solvers
A low-dissipative solver for turbulent compressible flows on unstructured meshes, with OpenFOAM implementation
Full text linkWe develop a high-fidelity numerical solver for the compressible Navier–Stokes equations, with the main aim of highlighting the predictive capabilities of low-diffusive numerics for flows in complex geometries. The space discretization of the convective terms in the Navier–Stokes equations relies on a robust energy-preserving numerical flux, and numerical diffusion inherited from the AUSM scheme is added limited to the vicinity of shock waves, or wherever spurious numerical oscillations are sensed. The solver is capable of conserving the total kinetic energy in the inviscid limit, and it bears sensibly less numerical diffusion than typical industrial solvers, with incurred greater predictive power, as demonstrated through a series of test cases including DNS, LES and URANS of turbulent flows. Simplicity of implementation in existing popular solvers such as OpenFOAM is also highlighted
An efficient semi-implicit solver for direct numerical simulation of compressible flows at all speeds
No full textWe develop a semi-implicit algorithm for time-accurate simulation of compressible
shock-free flows, with special reference to wall-bounded flows. The method is based on
partial linearization of the convective fluxes in such a way to suppress, or at least mitigate the
acoustic time step limitation. Together with replacement of the total energy equation with
the entropy transport equation, this approach avoids the inversion of block-banded matrices
involved in classical methods, which is replaced by much less demanding inversion of
standard banded matrices. The method is extended to deal with implicit integration of viscous
terms and to multiple space dimensions through approximate factorization, and used
as a building block of a semi-implicit Runge–Kutta scheme which guarantees third-order
of accuracy in time (Nikitin in Int J Numer Methods Fluids 51:221–233, 2006). Numerical
experiments are carried out for isotropic turbulence, plane channel flow, and flow in a square
duct. All available data support higher computational efficiency than existing methods, and
saving of computer resources ranging from 85% under low-subsonic flow conditions (down
to M0 ∼ 0.1), to about 50% in supersonic flow
DNS of turbulent flows in ducts with omplex shape
No full textWe carry out Direct Numerical Simulation (DNS) of flows in closed straight ducts with complex peripheral shape. To perform the simulations the Navier-Stokes equations in cylindrical coordinates are discretized by a second-order finite difference scheme, and the immersed-boundary technique is used to resolve the flow close to walls of complex shape. The basic geometry is a circular pipe of radius R, with imposed sinusoidal perturbations of the type ηRsin (Nw). Simulations by varying Nwat fixed η were performed to investigate the effect of the perturbation wavenumber. Additional simulations by fixing Nwand varying η also allow to investigate the influence of the amplitude of the wall corrugations. The modifications of the near-wall structures due to change in the shape of the walls are well depicted through contour plots of the radial component of the vorticity. The presence of geometrical disturbances anchors the structures at the locations where curvature changes, and the shape of the structures is strongly linked to the amplitude of the wall corrugation. Our interest is also in understanding the influence of the shape of the surface on wall friction. We were expecting some changes in the profile of the total stress with respect to that of the circular pipe, which instead were not found. This is a first indication that changes in the near-wall region do not affect the outer region, and that Townsend’s similarity hypothesis holds
High-Reynolds-number effects on turbulent scalings in compressible channel flow
Full text linkThe effect of the Reynolds number in a supersonic isothermal channel flow is studied using a direct numerical simulation (DNS). The bulk Mach number based on the wall temperature is 1.5, and the bulk Reynolds number is increased up to Reτ ≈ 1000. The use of van Driest velocity transformation in the presence of heated walls has been questioned due to the poor accuracy at low Reynolds number. For this reason alternative transformations of the velocity profile and turbulence statistics have been proposed, as, for instance, semi-local scalings. We show that the van Driest transformation recovers its accuracy as the Reynolds number is increased. The Reynolds stresses collapse on the incompressible ones, when properly scaled with density, and very good agreement with the incompressible stresses is found in the outer layer
Turbulence and secondary motions in square duct flow
Full text linkWe study turbulent flows in pressure-driven ducts with square cross-section through
direct numerical simulation in a wide enough range of Reynolds number to reach
flow conditions which are representative of fully developed turbulence (Re 1000).
Numerical simulations are carried out over very long integration times to get
adequate convergence of the flow statistics, and specifically to achieve high-fidelity
representation of the secondary motions which arise. The intensity of the latter
is found to be on the order of 1 %–2% of the bulk velocity, and approximately
unaffected by Reynolds number variation, at least in the range under scrutiny. The
smallness of the mean convection terms in the streamwise vorticity equation points
to a simple characterization of the secondary flows, which in the asymptotic high-Re
regime are approximated with good accuracy by eigenfunctions of the Laplace
operator, in the core part of the duct. Despite their effect of redistributing the wall
shear stress along the duct perimeter, we find that secondary motions do not have
a large influence on the bulk flow properties, and the streamwise velocity field can
be characterized with good accuracy as resulting from the superposition of four flat
walls in isolation. As a consequence, we find that parametrizations based on the
hydraulic diameter concept, and modifications thereof, are successful in predicting the
duct friction coefficient
On the role of secondary motions in turbulent square duct flow
No full textWe use a direct numerical simulations (DNS) database for turbulent flow in a square duct up to bulk Reynolds number Reb = 40 000 to quantitatively analyse the role of secondary motions on the mean flow structure. For that purpose we derive a generalized form of the identity of Fukagata, Iwamoto and Kasagi (FIK), which allows one to quantify the effect of cross-stream convection on the mean streamwise velocity, wall shear stress and bulk friction coefficient. Secondary motions are found to contribute approximately 6% of the total friction, and to act as a self-regulating mechanism of turbulence whereby wall shear stress non-uniformities induced by corners are equalized, and universality of the wall-normal velocity profiles is established. We also carry out numerical experiments whereby the secondary motions are artificially suppressed, in which case their equalizing role is partially taken by the turbulent stresses
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