The formation of shear and density layers in stably stratified turbulent flows: linear processes

2002 ◽  
Vol 455 ◽  
pp. 243-262 ◽  
Author(s):  
M. GALMICHE ◽  
J. C. R. HUNT
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Shear Stresses ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
Linear Processes ◽  
Buoyancy Forces ◽  
Uniform Background ◽  
The Mean

The initial evolution of the momentum and buoyancy fluxes in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s. velocity u′0 and integral lengthscale l0 is calculated using a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) in order to study the growth of small temporal and spatial perturbations in the large-scale mean stratification N(z, t) and mean velocity profile ū(z, t) (here N is the local Brunt–Väisälä frequency and ū is the local velocity of the horizontal mean flow) when the ratio of buoyancy forces to inertial forces is large, i.e. Nl0/u′0[Gt ]1. The lengthscale L of the perturbations in the mean profiles of stratification and shear is assumed to be large compared to l0 and the presence of a uniform background mean shear can be taken into account in the model provided that the inertial shear forces are still weaker than the buoyancy forces, i.e. when the Richardson number Ri = (N/∂zū)2[Gt ]1 at each height.When a mean shear perturbation is introduced initially with no uniform background mean shear and uniform stratification, the analysis shows that the perturbations in the mean flow profile grow on a timescale of order N-1. When the mean density profile is perturbed initially in the absence of a background mean shear, layers with significant density gradient fluctuations grow on a timescale of order N−10 (where N0 is the order of magnitude of the initial Brunt–Väisälä frequency) without any associated mean velocity gradients in the layers. These results are in good agreement with the direct numerical simulations performed by Galmiche et al. (2002) and are consistent with the earlier physically based conjectures made by Phillips (1972) and Posmentier (1977). The model also shows that when there is a background mean shear in combination with perturbations in the mean stratification, negative shear stresses develop which cause the mean velocity gradient to grow in the density layers. The linear analysis for short times indicates that the scale on which the mean perturbations grow fastest is of order u′0/N0, which is consistent with the experiments of Park et al. (1994).We conclude that linear mechanisms are widely involved in the formation of shear and density layers in stratified flows as is observed in some laboratory experiments and geophysical flows, but note that the layers are also significantly influenced by nonlinear and dissipative processes at large times.

scholarly journals Simulations of rib-roughened rough-to-smooth turbulent channel flows

2018 ◽  
Vol 843 ◽  
pp. 419-449 ◽  
Author(s):  
Umair Ismail ◽  
Tamer A. Zaki ◽  
Paul A. Durbin
Keyword(s):  
Large Scale ◽  
Wall Layer ◽  
Shear Stresses ◽  
Thin Wall ◽  
Computational Domain ◽  
Reynolds Stresses ◽  
Smooth Wall ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

High-fidelity simulations of turbulent flow through a channel with a rough wall, followed by a smooth wall, demonstrate a high degree of non-equilibrium within the recovery region. In fact, the recovery of all the flow statistics studied is incomplete by the streamwise exit of the computational domain. Above a thin wall layer, turbulence intensities significantly higher than fully developed, smooth-wall levels persist in the developing region. Within the thin wall layer, the profile shapes for turbulence stresses recover very quickly and wall-normal locations of characteristic peaks are established. However, even in this thin layer, complete recovery of magnitudes of turbulence stresses is exceptionally slow. A similar initially swift but eventually incomplete and slow relaxation behaviour is also shown by the skin friction. Between the turbulence shear and streamwise stresses, the turbulence shear stress shows a comparatively quick rate of recovery above a thin wall layer. Over the developing smooth wall, the balance is not merely between fluxes due to pressure and shear stresses. Strong momentum fluxes, which are directly influenced by the upstream roughness size, contribute significantly to this balance. Approximate curve fits estimate the streamwise distance required by the outer peaks of Reynolds stresses to attain near-fully-developed levels at approximately $20\unicode[STIX]{x1D6FF}{-}25\unicode[STIX]{x1D6FF}$, with $\unicode[STIX]{x1D6FF}$ being the channel half height. An even longer distance, of more than $50\unicode[STIX]{x1D6FF}$, might be needed by the mean velocity to approach near-fully-developed magnitudes. Visualizations and correlations show that large-scale eddies that are created above the roughness persist downstream, and sporadically perturb the elongated streaks. These streaks of alternating high and low momentum appear almost instantly after the roughness is removed. The mean flow does not re-establish an equilibrium log layer within the computational domain, and the velocity deficit created by the roughness continues throughout the domain. On the step change in roughness, near the wall, profiles for turbulence kinetic energy dissipation rate, $\unicode[STIX]{x1D716}$, and energy spectra indicate a sharp reduction in energy at small scales. Despite this, reversion towards equilibrium smooth-wall levels is slow, and ultimately incomplete, due to a rather slow adjustment of the turbulence cascade. The non-dimensional roughness height, $k^{+}$ ranges from 42 to 254 and the friction velocity Reynolds number at the smooth wall, $Re_{\unicode[STIX]{x1D70F}S}$, ranges from 284 to 1160 in the various simulations.

The structure of a highly decelerated axisymmetric turbulent boundary layer

2021 ◽  
Vol 929 ◽  
Author(s):  
N. Agastya Balantrapu ◽  
Christopher Hickling ◽  
W. Nathan Alexander ◽  
William Devenport
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Shear Layer ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Large Scale ◽  
Convection Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

Experiments were performed over a body of revolution at a length-based Reynolds number of 1.9 million. While the lateral curvature parameters are moderate ( $\delta /r_s < 2, r_s^+>500$ , where $\delta$ is the boundary layer thickness and r s is the radius of curvature), the pressure gradient is increasingly adverse ( $\beta _{C} \in [5 \text {--} 18]$ where $\beta_{C}$ is Clauser’s pressure gradient parameter), representative of vehicle-relevant conditions. The mean flow in the outer regions of this fully attached boundary layer displays some properties of a free-shear layer, with the mean-velocity and turbulence intensity profiles attaining self-similarity with the ‘embedded shear layer’ scaling (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642). Spectral analysis of the streamwise turbulence revealed that, as the mean flow decelerates, the large-scale motions energize across the boundary layer, growing proportionally with the boundary layer thickness. When scaled with the shear layer parameters, the distribution of the energy in the low-frequency region is approximately self-similar, emphasizing the role of the embedded shear layer in the large-scale motions. The correlation structure of the boundary layer is discussed at length to supply information towards the development of turbulence and aeroacoustic models. One major finding is that the estimation of integral turbulence length scales from single-point measurements, via Taylor's hypothesis, requires significant corrections to the convection velocity in the inner 50 % of the boundary layer. The apparent convection velocity (estimated from the ratio of integral length scale to the time scale), is approximately 40 % greater than the local mean velocity, suggesting the turbulence is convected much faster than previously thought. Closer to the wall even higher corrections are required.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

scholarly journals WKB theory for rapid distortion of inhomogeneous turbulence

1999 ◽  
Vol 390 ◽  
pp. 325-348 ◽  
Author(s):  
S. NAZARENKO ◽  
N. K.-R. KEVLAHAN ◽  
B. DUBRULLE
Keyword(s):  
Large Scale ◽  
Isotropic Turbulence ◽  
Two Dimensional ◽  
Turbulent Spot ◽  
Mean Flow ◽  
Mean Velocity ◽  
Inhomogeneous Turbulence ◽  
Large Scale Vortex ◽  
The Mean ◽  
Mean Strain

A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

scholarly journals A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow

2015 ◽  
Vol 774 ◽  
pp. 143-169 ◽  
Author(s):  
J.-H. Xie ◽  
J. Vanneste
Keyword(s):  
Large Scale ◽  
Interaction Mechanism ◽  
Coupled Model ◽  
Inertial Waves ◽  
Mean Flow ◽  
Mean Velocity ◽  
Time Scale Separation ◽  
The Mean ◽  
Balanced Flow ◽  
Mesoscale Motion

Wind forcing of the ocean generates a spectrum of inertia–gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) are highly energetic and play a significant role in the slow, large-scale dynamics of the ocean. To analyse this role, we develop a new model of the non-dissipative interactions between NIWs and balanced motion. The model is derived using the generalised-Lagrangian-mean (GLM) framework (specifically, the ‘glm’ variant of Soward & Roberts, J. Fluid Mech., vol. 661, 2010, pp. 45–72), taking advantage of the time-scale separation between the two types of motion to average over the short NIW period. We combine Salmon’s (J. Fluid Mech., vol. 719, 2013, pp. 165–182) variational formulation of GLM with Whitham averaging to obtain a system of equations governing the joint evolution of NIWs and mean flow. Assuming that the mean flow is geostrophically balanced reduces this system to a simple model coupling Young & Ben Jelloul’s (J. Mar. Res., vol. 55, 1997, pp. 735–766) equation for NIWs with a modified quasi-geostrophic (QG) equation. In this coupled model, the mean flow affects the NIWs through advection and refraction; conversely, the NIWs affect the mean flow by modifying the potential-vorticity (PV) inversion – the relation between advected PV and advecting mean velocity – through a quadratic wave term, consistent with the GLM results of Bühler & McIntyre (J. Fluid Mech., vol. 354, 1998, pp. 301–343). The coupled model is Hamiltonian and its conservation laws, for wave action and energy in particular, prove illuminating: on their basis, we identify a new interaction mechanism whereby NIWs forced at large scales extract energy from the balanced flow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. A rough estimate suggests that this mechanism could provide a significant sink of energy for mesoscale motion and play a part in the global energetics of the ocean. Idealised two-dimensional models are derived and simulated numerically to gain insight into NIW–mean-flow interaction processes. A simulation of a one-dimensional barotropic jet demonstrates how NIWs forced by wind slow down the jet as they propagate into the ocean interior. A simulation assuming plane travelling NIWs in the vertical shows how a vortex dipole is deflected by NIWs, illustrating the irreversible nature of the interactions. In both simulations energy is transferred from the mean flow to the NIWs.

Structure of Turbulent Flows Over Forward Facing Steps With Adverse Pressure Gradient

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Hassan Iftekhar ◽  
Martin Agelin-Chaab
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Turbulent Flows ◽  
Large Scale ◽  
Adverse Pressure Gradient ◽  
Reynolds Numbers ◽  
Step Height ◽  
Reattachment Length ◽  
Mean Velocity ◽  
The Mean

This paper reports an experimental study on the effects of adverse pressure gradient (APG) and Reynolds number on turbulent flows over a forward facing step (FFS) by employing three APGs and three Reynolds numbers. A particle image velocimetry (PIV) technique was used to conduct velocity measurements at several locations downstream, and the flow statistics up to 68 step heights are reported. The step height was maintained at 6 mm, and the Reynolds numbers based on the step height and freestream mean velocity were 1600, 3200, and 4800. The mean reattachment length increases with the increase in Reynolds number without the APG whereas the mean reattachment length remains constant for increasing APG. The proper orthogonal decomposition (POD) results confirmed that higher Reynolds numbers caused the large-scale structures to be more defined and organized close to the step surface.

Large-scale features in turbulent pipe and channel flows

2007 ◽  
Vol 589 ◽  
pp. 147-156 ◽  
Author(s):  
J. P. MONTY ◽  
J. A. STEWART ◽  
R. C. WILLIAMS ◽  
M. S. CHONG
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Turbulent Flows ◽  
Large Scale ◽  
Significant Progress ◽  
Mean Velocity ◽  
Characteristic Structure ◽  
Custom Made ◽  
Logarithmic Region ◽  
The Mean

In recent years there has been significant progress made towards understanding the large-scale structure of wall-bounded shear flows. Most of this work has been conducted with turbulent boundary layers, leaving scope for further work in pipes and channels. In this article the structure of fully developed turbulent pipe and channel flow has been studied using custom-made arrays of hot-wire probes. Results reveal long meandering structures of length up to 25 pipe radii or channel half-heights. These appear to be qualitatively similar to those reported in the log region of a turbulent boundary layer. However, for the channel case, large-scale coherence persists further from the wall than in boundary layers. This is expected since these large-scale features are a property of the logarithmic region of the mean velocity profile in boundary layers and it is well-known that the mean velocity in a channel remains very close to the log law much further from the wall. Further comparison of the three turbulent flows shows that the characteristic structure width in the logarithmic region of a boundary layer is at least 1.6 times smaller than that in a pipe or channel.

Correlation measurements in a non-frozen pattern of turbulence

1964 ◽  
Vol 18 (1) ◽  
pp. 97-116 ◽  
Author(s):  
M. J. Fisher ◽  
P. O. A. L. Davies
Keyword(s):  
Convection Velocity ◽  
Wave Number ◽  
Shear Stresses ◽  
Mean Flow ◽  
Velocity Correlation ◽  
Mean Velocity ◽  
Space Correlation ◽  
Spectral Components ◽  
Small Wave Number ◽  
The Mean

The properties of a turbulent flow are often described in terms of velocity correlations in space, in time, and in space-time. In this paper the interpretation of velocity correlation measurements which are made in a region of highintensity turbulence is considered in some detail. Under these conditions it is shown that some account must be taken of the effects of both mean and fluctuating shear stresses which are continuously modifying the turbulent structure. For an almost frozen pattern, for example, in the turbulence behind a grid, the turbulent convection velocity is amost equal to the mean flow velocity, while the space correlation and auto-correlation of the velocity fluctuations are simply related through this velocity. In contrast to this, when the intensity is high, the convection velocity may differ considerably from the mean velocity, while it is shown that different turbulent spectral components appear to travel at different speeds. This means that the turbulent spectrum and the turbulent space scales are no longer simply related. For example, the high-frequency spectral components may be ascribed to both the high-velocity eddies and the small wave-number components acting together.Experimental results are presented which indicate the conditions under which the assumption of a frozen pattern leads to uncertainties in the subsequent interpretation of the measurements. The measurements also show that the observed difference between the mean and the convection velocity may be qualitatively explained in terms of the skewness of the velocity signals.

On Coupling of RANS and LES for Integrated Computations of Jet Engines

2007 ◽  
Author(s):  
Gorazd Medic ◽  
Donghyun You ◽  
Georgi Kalitzin
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Flow Simulation ◽  
Operating Conditions ◽  
Jet Engines ◽  
Mean Flow ◽  
Mean Velocity ◽  
Novel Approach ◽  
The Mean ◽  
Flow Variables

Large scale integrated computations of jet engines can be performed by using the unsteady RANS framework to compute the flow in turbomachinery components while using the LES framework to compute the flow in the combustor. This requires a proper coupling of the flow variables at the interfaces between the RANS and LES solvers. In this paper, a novel approach to turbulence coupling is proposed. It is based on the observation that in full operating conditions the mean flow at the interfaces is highly non-uniform and local turbulence production dominates convection effects in regions of large velocity gradients. This observation has lead to the concept of using auxilliary ducts to compute turbulence based on the mean velocity at the interface. In the case of the RANS/LES interface, turbulent fluctuations are reconstructed from an LES computation in an auxiliary three-dimensional duct using a recycling technique. For the LES/RANS interface, the turbulence variables for the RANS model are computed from an auxilliary solution of the RANS turbulence model in a quasi-2D duct. We have demonstrated the feasibility of this approach for the integrated flow simulation of a 20° sector of an entire jet engine.

Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction

2014 ◽  
Vol 745 ◽  
pp. 378-397 ◽  
Author(s):  
Sergio Pirozzoli
Keyword(s):  
Velocity Profile ◽  
Turbulent Flows ◽  
Outer Part ◽  
Wall Layer ◽  
Wall Friction ◽  
Mixing Length ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
Turbulent Wall

AbstractWe reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

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