Showing papers in "Theoretical and Computational Fluid Dynamics in 2006"

A new version of detached-eddy simulation, resistant to ambiguous grid densities

Abstract: Detached-eddy simulation (DES) is well understood in thin boundary layers, with the turbulence model in its Reynolds-averaged Navier–Stokes (RANS) mode and flattened grid cells, and in regions of massive separation, with the turbulence model in its large-eddy simulation (LES) mode and grid cells close to isotropic. However its initial formulation, denoted DES97 from here on, can exhibit an incorrect behavior in thick boundary layers and shallow separation regions. This behavior begins when the grid spacing parallel to the wall Δ∥ becomes less than the boundary-layer thickness δ, either through grid refinement or boundary-layer thickening. The grid spacing is then fine enough for the DES length scale to follow the LES branch (and therefore lower the eddy viscosity below the RANS level), but resolved Reynolds stresses deriving from velocity fluctuations (“LES content”) have not replaced the modeled Reynolds stresses. LES content may be lacking because the resolution is not fine enough to fully support it, and/or because of delays in its generation by instabilities. The depleted stresses reduce the skin friction, which can lead to premature separation.

MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel

Abstract: Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.

Investigation of downstream and sideline subsonic jet noise using Large Eddy Simulation

Abstract: The sound fields radiated by Mach number 0.6 and 0.9, circular jets with Reynolds numbers varying from 1.7×103 to 4×105 are investigated using Large Eddy Simulations. As the Reynolds number decreases, the properties of the sound radiation do not change significantly in the downstream direction, whereas they are modified in the sideline direction. At low Reynolds numbers, for large angles downstream from the jet axis, the acoustic levels are indeed remarkably lower and a large high-frequency part of the sound spectra vanishes. For all Reynolds numbers, the downstream and the sideline sound spectra both appear to scale in frequency with the Strouhal number. However their peak amplitudes vary following two different velocity exponents according to the radiation direction. The present observations suggest the presence of two sound sources: a Reynolds number-dependent source, predominant for large radiation angles, connected to the randomly-developing turbulence, and a deterministic source, radiating downstream, related to a mechanism intrinsic to the jet geometry, which is still to be comprehensively described. This view agrees well with the experimental results displaying two distinguishable components in turbulent mixing noise [1, 2].

Bulk boundary-layer concepts for simplified models of tropical dynamics

Abstract: We review bulk representations of tropical and subtropical maritime atmospheric boundary layers. Three types of bulk representations are studied in detail: stratocumulus topped mixed layers, trade-wind layers, and sub-cloud mixed layers. Through the development of a consistent description of these disparate regimes, connections among their varied representations are emphasized, as well as their relation to regions of deeper convection. New results relating to the equilibrium mass flux and cloud fraction in the trade winds; the ability of bulk models to represent qualitatively major cloud regimes; and the relationship amongst different bulk representations of the surface layer are presented. Throughout we emphasize the identification of consistent and physically based mixing and cloud regime rules for use in intermediate complexity models of the tropical climate, which in turn can be used to study cloud and dynamical interactions on larger scales.

The boundary layer contribution to intertropical convergence zones in the quasi-equilibrium tropical circulation model framework

Abstract: Theories for the position and intensity of precipitation over tropical oceans on climate time scales have a perplexing disagreement between those that focus on the momentum budget of the atmospheric boundary layer (ABL) and those that focus on thermodynamic factors. In the case of narrow intertropical convergence zones (ITCZs), there is some evidence for both classes of theories, and there are large open questions on the interpretation of the moist static energy (MSE) and momentum budgets of these regions. We develop a model in which both types of mechanisms can operate and the interaction between them can be analyzed. The model includes a mixed-layer ABL, coupled to a free troposphere whose vertical structure follows the quasi-equilibrium tropical circulation model (QTCM) of Neelin and Zeng. The case analyzed here is axisymmetric, using a fixed sea surface temperature (SST) lower boundary condition with an idealized off-equatorial SST maximum. We examine a regime with small values of the gross moist stability associated with tropospheric motions, which is realistic but poses theoretical challenges. In both rotating (equatorial β-plane) and nonrotating cases, the model ITCZ width and intensity are substantially controlled by the horizontal diffusion of moisture, which is hypothesized to be standing in for nonaxisymmetric transients. The inclusion of the ABL increases the amplitude and sharpness of the ITCZ, contributing to the importance of diffusion. Analytical solutions under simplifying assumptions show that the ABL contribution is not singular in the nondiffusive limit; it just features an ITCZ more intense than observed. A negative gross moist stability contribution associated with the flow component driven by ABL momentum dynamics plays a large role in this. Because of the ABL contribution, the flow imports, rather than exports, MSE in the ITCZ, but we show that this can be understood rather simply. The ABL contribution can be approximately viewed as a forcing to the tropospheric thermodynamics. The ABL forcing term is in addition to thermodynamic forcing by net flux terms in the MSE budget, which otherwise is much as in the standard QTCM. The ABL momentum budget suggests that divergent flow in the ABL is controlled to a significant extent by the pressure gradient imprinted on the ABL by the SST gradient—termed the Lindzen–Nigam contribution—although we also find that the thermodynamics mediating this is nontrivial, especially in the rotating case. Nonetheless, when this component of the pressure gradient is artificially removed, the peak ITCZ precipitation is reduced by a fraction on the order of 15 to 25%, less than might have been expected based on the diagnosis of the ABL momentum budget.

Systematic multiscale models for deep convection on mesoscales

Abstract: This paper builds on recent developments of a unified asymptotic approach to meteorological modelling (Klein (2000), Klein (2003)), which was used successfully in the development of “Systematicmultiscale models for the tropics” in (Majda & Klein (2003), Majda & Biello (2004), Biello & Majda(2005)). Here we account for typical bulk microphysics parameterizations of moist processes within this framework. The key steps are careful nondimensionalization of the bulk microphysics equations and the choice of appropriate distinguished limits for the various nondimensional small parameters that appear. We are then in the position to study scale interactions in the atmosphere involving moist physics. We demonstrate this by developing two systematic multiscale models that are motivated by our interest in mesoscale organized convection. The emphasis here is on multiple length, but common time scales. The first of these models describes the short time evolution of slender, deep convective “hot towers” with horizontal scale 1 km interacting with the linearized momentum balance on length and time scales of (10km / 3 min). We expect this model to describe how convective inhibition may be overcome near the surface, how the onset of deep convection triggers convective scale gravity waves, and that it will also yield new insight into how such local convective events may conspire to create larger scale strong storms. The second model addresses the next larger range of length and time scales (10 km, 100 km, and 20min) and exhibits mathematical features that are strongly reminiscent of mesoscale organized convection. In both cases, the asymptotic analysis reveals how the stiffness of condensation/evaporation processes induces highly nonlinear dynamics. Besides providing new theoretical insights, the derived models may also serve as a theoretical devices for analyzing and interpreting the results of complex moist process model simulations, and they may stimulate the development of new, theoretically grounded subgrid scale parameterizations.

Multicloud convective parametrizations with crude vertical structure

Abstract: Recent observational analysis reveals the central role of three multi-cloud types, congestus, stratiform, and deep convective cumulus clouds, in the dynamics of large scale convectively coupled Kelvin waves, westward propagating two-day waves, and the Madden–Julian oscillation. The authors have recently developed a systematic model convective parametrization highlighting the dynamic role of the three cloud types through two baroclinic modes of vertical structure: a deep convective heating mode and a second mode with low level heating and cooling corresponding respectively to congestus and stratiform clouds. The model includes a systematic moisture equation where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep convective precipitation and a nonlinear switch which favors either deep or congestus convection depending on whether the troposphere is moist or dry. Here several new facets of these multi-cloud models are discussed including all the relevant time scales in the models and the links with simpler parametrizations involving only a single baroclinic mode in various limiting regimes. One of the new phenomena in the multi-cloud models is the existence of suitable unstable radiative convective equilibria (RCE) involving a larger fraction of congestus clouds and a smaller fraction of deep convective clouds. Novel aspects of the linear and nonlinear stability of such unstable RCE’s are studied here. They include new modes of linear instability including mesoscale second baroclinic moist gravity waves, slow moving mesoscale modes resembling squall lines, and large scale standing modes. The nonlinear instability of unstable RCE’s to homogeneous perturbations is studied with three different types of nonlinear dynamics occurring which involve adjustment to a steady deep convective RCE, periodic oscillation, and even heteroclinic chaos in suitable parameter regimes.

Direct numerical simulation of disturbances generated by periodic suction-blowing in a hypersonic boundary layer

Abstract: A numerical algorithm and code are developed and applied to direct numerical simulation (DNS) of unsteady two-dimensional flow fields relevant to stability of the hypersonic boundary layer. An implicit second-order finite-volume technique is used for solving the compressible Navier–Stokes equations. Numerical simulation of disturbances generated by a periodic suction-blowing on a flat plate is performed at free-stream Mach number 6. For small forcing amplitudes, the second-mode growth rates predicted by DNS agree well with the growth rates resulted from the linear stability theory (LST) including nonparallel effects. This shows that numerical method allows for simulation of unstable processes despite its dissipative features. Calculations at large forcing amplitudes illustrate nonlinear dynamics of the disturbance flow field. DNS predicts a nonlinear saturation of fundamental harmonic and rapid growth of higher harmonics. These results are consistent with the experimental data of Stetson and Kimmel obtained on a sharp cone at the free-stream Mach number 8.

Ligament formation in sheared liquid-gas layers

Abstract: We perform numerical simulations of two-phase liquid–gas sheared layers, with the objective of studying atomization. The Navier–Stokes equations for two-dimensional incompressible flow are solved in a periodic domain. A volume-of-fluid method is used to track the interface. The density ratio is kept around 10. The calculations show good agreement with a fully viscous Orr–Sommerfeld linear theory over several orders of magnitude of interface growth. The nonlinear development shows the growth of finger-like structures, or ligaments, and the detachment of droplets. The effect of the Weber and Reynolds numbers, the boundary layer width and the initial perturbation amplitude are discussed through a number of typical cases. Inversion of the liquid boundary layer is shown to yield more readily ligaments bending upwards and is thus more likely to produce droplets.

A simple equilibrium model for shallow-cumulus-topped mixed layers

Abstract: A new equilibrium model for shallow-cumulus-topped mixed layers is presented. A variant on the w∗ closure for the shallow-cumulus mass flux is applied that retains the convective area-fraction in its formu- lation. As opposed to being constant, the fraction is explicitly modeled using a statistical closure as a function of the saturation deficit and humidity variance at cloud base. As a consequence, important new interactions are introduced between the convective transport, humidity, and depth of the mixed layer. This mechanism, which we call the mass-flux humidity feedback, helps determine the character of the equilibrium state such that the mixed-layer top is maintained close to the cloud-base height. Due to the strong sensitivity of the mass flux to the area fraction, the latter thus acts as a regulator or valve mechanism on moist convective transport. As a consequence, the mixed-layer model is able to explain the robustness of many aspects of the shallow-cumulus boundary layer that is typically found in observations and large-eddy simulations (LESs). The model is evaluated for a single-LES case as well as for global climatology obtained from a 40-year reanalysis of meteorological data by the European Centre for Medium-range Weather Forecasts (ECMWF). LES characteristics of convective mass flux, cloud fraction, humidity variance, cloud-base height, and surface fluxes of heat and humidity are reproduced. The solution on reanalysis fields reproduces the spatial structure of mixed-layer temperature and humidity and their associated surface fluxes in the subtropical Atlantic and Pacific trade wind regions. Furthermore, the spatial structure of the convective area-fraction matches that of synoptic surface observations of frequency of occurrence of shallow cumulus. Particularly striking is the smooth onset of the convective area-fraction and mass flux along the trade-wind trajectory that is reproduced, from zero to typical trade-wind values. The cumulus onset represents the necessity for shallow-cumulus mass flux to occur in order to close the mixed-layer budgets of heat, moisture, and mass, as a response to the changing magnitude of large-scale subsidence and free tropospheric humidity along the trajectory. Finally, the mass flux model is implemented in an intermediate-complexity tropical climate model to study its behavior when fully interactive with the larger-scale flow. A climate run then shows that the model is stable, due to the mass-flux humidity feedback acting to keep the shallow-cumulus boundary layer close to its equilibrium state for long, climatological timescales.

Structure of tropical variability from a vertical mode perspective

Abstract: A composite mesoscale precipitation event and a convectively coupled Kelvin wave produced by a diabatically accelerated cloud resolving model are compared. Special emphasis is placed on the vertical structure of density and moisture perturbations and the interaction of these perturbations with the composited dynamical fields. Both composites share the same general features, a gradual deepening and strengthening of convection followed by deep convection and a stratiform region, quite similar in character to observations and some recent idealized models. Composited frozen moist static energy (FMSE) perturbations are several times larger than virtual temperature perturbations, suggesting moisture is a dominant regulator of convection. An empirically derived two vertical mode decomposition of the dynamical and moisture fields is found to reproduce both composites quite well. The leading vertical modes of FMSE and virtual temperature variability are strongly correlated with the modes of vertical velocity variability; these correlations are strongest at near-zero time lags. Deep convection is associated with moistening in the lower and middle troposphere, while shallow convection is associated with a moist lower troposphere and dry middle and upper troposphere. To the extent that our numerical model is realistic, the empirical modal decomposition provides support for the use of two-mode idealized models for convective interaction with large-scale circulations and guidance for formulating feedbacks between convection and the thermodynamic profile in such models. The FMSE budget leads to an interpretation of the convective life-cycle as a recharge–discharge mechanism in column-integrated FMSE. The budget analysis places diabatic forcing, surface and radiative fluxes into the moist energetic framework. In particular, these fluxes are seen to prolong active convection, but play a passive role in its initiation. The modally decomposed FMSE budget highlights the dynamical importance of the second baroclinic mode in moistening the lower and middle troposphere before convective onset (recharging), and then discharging stored FMSE in the stratiform region.

Interpretation of simple and cloud-resolving simulations of moist convection radiation interaction with a mock-Walker circulation

Abstract: An idealized two-dimensional mock-Walker circulation in the tropical atmosphere forced by prescribed horizontal gradients in sea-surface temperature (SST) is discussed. This model problem includes feedbacks between cumulus convection and tropical large-scale circulations that have proved challenging for global climate models to predict accurately. Three-dimensional cloud-resolving model (CRM) simulations that explicitly simulate turbulent circulations within individual cloud systems across 4,096 and 1,024 km-wide Walker circulations are compared with a simple theoretical model, the Simplified Quasiequilibrium Tropical Circulation Model (SQTCM). This theoretical model combines the weak-temperature-gradient approximation with a unimodal truncation of tropospheric vertical structure coupled to highly simplified formulations of moist precipitating cumulus convection and its cloud-radiative feedbacks. The rainfall, cloud and humidity distribution, circulation strength, energy fluxes and scaling properties are compared between the models. The CRM-simulated horizontal distribution of rainfall and energy fluxes are adequately predicted by the SQTCM. However, the humidity distribution (drier subsidence regions and high-humidity boundary layers in the CRM), vertical structure and domain-size scaling of the circulation differ significantly between the models. For the SQTCM, the concept of gross moist stability – related to advection of moist static energy (MSE) out of tropospheric columns by the mean divergent circulation – is used to explain the width and intensity of the rainy region. Column MSE budgets averaged across the ascent branch of the simulated Walker circulation provide similar insight into the cloud-resolving simulations after consideration of the more complex horizontal and vertical circulation structure and the role of transient eddies. A nondimensional ascent-region moist stability ratio α, analogous to the SQTCM gross moist stability, is developed. One term of α is related to the vertical profiles of ascent-region mean vertical motion and ascent-region edge MSE; a second term is proportional to eddy export from the ascent region. Smaller α induces a narrower, rainier ascent region. The sensitivity of the SQTCM and CRM to a uniform 2 K increase in SST is compared, and the rainy upward branch of the circulation narrows in both models. MSE budget arguments are used to explain this behavior. In the simple model, the gross moist stability is a decreasing function of tropospheric temperature. Hence gross moist stability reduces and the ascent region narrows as the SST increases. In the CRM, increased atmospheric radiative cooling due to the warmer and moister troposphere destabilizes the MSE profile and decreases α, inducing a narrower ascent region. In the CRM, and to a lesser extent in the SQTCM, intensified shortwave cloud forcing in the warmer climate causes a negative radiative feedback on the SST change.

Steady Free-surface Flow Past an Uneven Channel Bottom

Abstract: New free-surface flows past a semi-infinite ‘step’ in the bottom of a channel are considered. Surface tension is neglected but gravity is included in the dynamic boundary condition. Fully nonlinear solutions are computed by boundary integral equation methods. Additional weakly nonlinear solutions are derived analytically. A thorough analysis of the weakly nonlinear problem provides a systematic approach to identify all the possible types of solutions and the number of independent parameters.

The Structure of Precipitation Fronts for Finite Relaxation Time

Abstract: When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper’s simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax’s stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.

A hybrid RANS/LES simulation of high-Reynolds-number channel flow using additional filtering at the interface

Abstract: A hybrid method combining large eddy simulation (LES) with the Reynolds-averaged Navier-Stokes (RANS) equation is used to simulate a turbulent channel flow at high Reynolds number. It is known that the mean velocity profile has a mismatch between the RANS and LES regions in hybrid simulations of a channel flow. The velocity mismatch is reproduced and its dependence on the location of the RANS/LES interface and on the type of RANS model is examined in order to better understand its properties. To remove the mismatch and to obtain better velocity profiles, additional filtering is applied to the velocity components in the wall-parallel planes near the interface. The additional filtering was previously introduced to simulate a channel flow at low Reynolds number. It is shown that the filtering is effective in reducing the mismatch even at high Reynolds number. Profiles of the velocity fluctuations of runs with and without the additional filtering are examined to help understand the reason for the mismatch. Due to the additional filtering, the wall-normal velocity fluctuation increases at the bottom of the LES region. The resulting velocity field creates the grid-scale shear stress more efficiently, and an overestimate of the velocity gradient is removed. The dependence of the velocity profile on the grid point number is also investigated. It is found that the velocity gradient in the core region is underestimated in the case of a coarse grid. Attention should be paid not only to the velocity mismatch near the interface but also to the velocity profile in the core region in hybrid simulations of a channel flow at high Reynolds number.

A non-linear SGS model based on the spatial velocity increment. Application to LES of fully developed pipe flow and round turbulent jet.

Abstract: A new subgrid scale (SGS) modelling concept for large-eddy simulation (LES) of incompressible flow is proposed based on the three-dimensional spatial velocity increment δ u i . The new model is inspired by the structure function formulation developed by Metais and Lesieur [39] and applied in the context of the scale similarity type formulation. First, the similarity between the SGS stress tensor τ ij and the velocity increment tensor Q ij = δ u i δ u j is analyzed analytically and numerically using a priori tests of fully developed pipe flow at Re τ = 180. Both forward and backward energy transfers between resolved and unresolved scales of the flow are well predicted with a SGS model based on Q ij . Secondly, a posteriori tests are performed for two families of turbulent shear flows. LES of fully developed pipe flow up to Re τ = 520 and LES of round turbulent jet at Re D = 25000 carried out with a dynamic version of the model provide promising results that confirm the power of this approach for wall-bounded and free shear flows.

A viscoelastic fluid model for describing the mechanics of a coarse ligated plasma clot

Abstract: Thrombi are formed at the end of a series of complex biochemical processes. There are various types of thrombi, and their rheological properties change depending on the conditions during clot formation. In this paper, a model for a particular type of clot, formed from human plasma, is proposed within a thermodynamic framework that recognizes that viscoelastic fluids possess multiple natural configurations.

Statistical properties of coherent fine eddies in wall-bounded turbulent flows by direct numerical simulation

Abstract: The characteristics of coherent fine eddies in wall-bounded turbulent flows are investigated by direct numerical simulation (DNS). The results show that when coherent eddies scaled with Kolmogorov microscale, η and root mean square (rms) of velocity fluctuations, u′rms, it is found that its average diameter is about 10η ∼ 12η and average maximum azimuthal velocity is about 0.5 ∼ 0.6u′rms. Mean azimuthal velocity of coherent eddies follows the profile of Burgers' vortex. Circulations of coherent vortices at different wall locations collapse in similar patterns and show power law behavior. A theoretical description of coherent eddies can be made based on the Burgers' vortex approximation. The diameter and maximum azimuthal velocity of coherent eddies are well approximated at different circulations of Burgers' vortices. It is observed that coherent eddies, those having an average diameter about 10η ∼ 12η, possess maximum azimuthal velocity and intensity ranges as far as 3u′rms. It is shown that the diameter and velocity of coherent eddies are strongly correlated. The simulation results may provide important insight into better understanding of the behaviors of coherent eddies in wall-bounded turbulent flows.

Numerical simulation of vortex ring formation in the presence of background flow with implications for squid propulsion

Abstract: Numerical simulations are used to study laminar vortex ring formation under the influence of background flow. The numerical setup includes a round-headed axisymmetric body with an opening at the posterior end from which a column of fluid is pushed out by a piston. The piston motion is explicitly included into the simulations by using a deforming mesh. A well-developed wake flow behind the body together with a finite-thickness boundary layer outside the opening is taken as the initial flow condition. As the jet is initiated, different vortex evolution behavior is observed depending on the combination of background flow velocity to mean piston velocity (\(U/U_{p}\)) ratio and piston stroke to opening diameter (\(L_{m}/D\)) ratio. For low background flow (\(U/U_{p} =0.2\)) with a short jet (\(L_{m}/D =6\)), a leading vortex ring pinches off from the generating jet, with an increased formation number. For intermediate background flow (\(U/U_{p} =0.5\)) with a short jet (\(L_{m}/D =6\)), a leading vortex ring also pinches off but with a reduced formation number. For intermediate background flow (\(U/U_{p} =0.5\)) with a long jet (\(L_{m}/D =15\)), no vortex ring pinch-off is observed. For high background flow (\(U/U_{p} =0.75\)) with both a short (\(L_{m}/D =6\)) and a long (\(L_{m}/D =15\)) jet, the leading vortex structure is highly deformed with no single central axis of fluid rotation (when viewed in cross-section) as would be expected for a roll-up vortex ring. For \(L_{m}/D =6\), the vortex structure becomes isolated as the trailing jet is destroyed by the opposite-signed vorticity of the background flow. For \(L_{m}/D =15\), the vortex structure never pinches off from the trailing jet. The underlying mechanism is the interaction between the vorticity layer of the jet and the opposite-signed vorticity layer from the initial wake. This interaction depends on both \(U/U_{p}\) and \(L_{m}/D\). A comparison is also made between the thrust generated by long, continuous jets and jet events constructed from a periodic series of short pulses having the same total mass flux. Force calculations suggest that long, continuous jets maximize thrust generation for a given amount of energy expended in creating the jet flow. The implications of the numerical results are discussed as they pertain to adult squid propulsion, which have been observed to generate long jets without a prominent leading vortex ring.

3D Steady and Unsteady Bifurcations in a Shock-wave/Laminar Boundary Layer Interaction: A Numerical Study

Abstract: The principal objective of this paper is to study some unsteady characteristics of an interaction between an incident oblique shock wave impinging a laminar boundary layer developing on a plate plane. More precisely, this paper shows that some unsteadiness, in particular the low frequency unsteadiness, originate in a supercritical Hopf bifurcation related to the dynamics of the separated boundary layer and not necessarily to the coherent structures resulting from the turbulent character of the boundary layer crossing the shock wave. Numerical computations of a shock-wave/laminar boundary-layer interaction (SWBLI) have been compared with a classical test case (Degrez test case) and both two-dimensional and three-dimensional (3D) unsteady Navier–Stokes equations are numerically solved with an implicit dual time stepping for the temporal algorithm and high order AUSM+ scheme for the spatial discretization. A parametric study on the oblique shock-wave angle has been performed to characterize the unsteady behaviour onset. Finally, discussions and assumptions are made about the origin of the 3D low frequency unsteadiness.

The hypohydrostatic rescaling and its impacts on modeling of atmospheric convection

Abstract: The atmospheric circulation spans a wide range of spatial scales, including the planetary scale (∼10,000 km), synoptic scale (∼2,000 km), mesoscale (∼200 km), and convective scales (<20 km). The wide scale separation between convective motions, responsible for the vertical energy transport, and the planetary circulation, responsible for the meridional energy transport, has prevented explicit representation of convective motions in global atmospheric models. Kuang et al. (Geophys. Res. Lett. 32: L02809, 2005) have suggested a way to circumvent this limitation through a rescaling that they refer to as Diabatic Acceleration and REscaling (DARE). We focus here on a modified version of the procedure that we refer to as hypohydrostatic rescaling. These two strategies are equivalent for inviscid and adiabatic flow in the traditional meteorological setting in which the vertical component of the Coriolis acceleration is ignored, but they differ when atmospheric physics is taken into account. It is argued here that, while the hypohydrostatic rescaling preserves the dynamics of the planetary scale circulation, it increases the horizontal scale of convective motions. This drastically reduces the computational cost for explicit simulation of hypohydrostatic convection in a global atmospheric model. A key question is whether explicit simulations of hypohydrostatic convection could offer a valid alternative to convective parameterization in global models. To do so, radiative-convective equilibrium is simulated with a high-resolution non-hydrostatic model using different model resolutions and values of the rescaling param- eter. When the behavior of hypohydrostatic convection is compared with coarse-resolution simulations of convection, the latter set of simulations reproduce more accurately the result from a reference high-resolution simulation. This is particularly true for the convective velocity and cloud ice distributions. Scaling arguments show that hypohydrostatic rescaling increases the convective overturning time. In particular, this convective slowdown associated with the hypohydrostatic rescaling is more significant than the slowdown resulting from

Predicting transport by Lagrangian coherent structures with a high-order method

Abstract: Recent developments in identifying Lagrangian coherent structures from finite-time velocity data have provided a theoretical basis for understanding chaotic transport in general flows with aperiodic dependence on time. As these theoretical developments are extended and applied to more complex flows, an accurate and general numerical method for computing these structures is needed to exploit these ideas for engineering applications. We present an unstructured high-order hp/spectral-element method for solving the two-dimensional compressible form of the Navier–Stokes equations. A corresponding high-order particle tracking method is also developed for extracting the Lagrangian coherent structures from the numerically computed velocity fields. Two different techniques are used; the first computes the direct Lyapunov exponent from an unstructured initial particle distribution, providing easier resolution of structures located close to physical boundaries, whereas the second advects a small material line initialized close to a Lagrangian saddle point to delineate these structures. We demonstrate our algorithm on simulations of a bluff-body flow at a Reynolds number of Re = 150 and a Mach number of M = 0.2 with and without flow forcing. We show that, in the unforced flow, periodic vortex shedding is predicted by our numerical simulations that is in stark contrast to the aperiodic flow field in the case with forcing. An analysis of the Lagrangian structures reveals a transport barrier that inhibits cross-wake transport in the unforced flow. The transport barrier is broken with forcing, producing enhanced transport properties by chaotic advection and consequently improved mixing of advected scalars within the wake.

Modeling Large-deformation-induced Microflow in Soft Biological Tissues

Abstract: The homogenization approach to multiscale modeling of soft biological tissues is presented. The homogenized model describes the relationship between the macroscopic hereditary creep behavior and the microflow in a fluid-saturated dual-porous medium at the microscopic level. The micromodel is based on Biot’s system for quasistatic deformation processes, modified for the updated Lagrangian formulation to account for coupling the fluid diffusion through a porous solid undergoing large deformation. Its microstructure is constituted by fluid-filled inclusions embedded in the porous matrix. The tangential stiffness coefficients and the retardation stress for the macromodel are derived for a time-stepping algorithm. Numerical examples are discussed, showing the strong potential of the model for simulations of deformation-driven physiological processes at the microscopic scale.

Investigation of the onset of bioconvection in a suspension of oxytactic microorganisms subjected to high-frequency vertical vibration

Abstract: This paper investigates the effect of vertical vibration on the stability of a dilute suspension of oxytactic microorganisms in a shallow horizontal fluid layer. For the case of high-frequency vibration, an averaging method is utilized to derive the equations describing the mean flow by decomposing the solutions of governing equations into two components: one that varies slowly with time, and a second that varies rapidly with time. Linear stability analysis is used to investigate the stability of the obtained averaged equations. It is predicted that high-frequency, low-amplitude vertical vibration has a stabilizing effect on a suspension of oxytactic microorganisms confined in a shallow horizontal layer.

Asymptotic solutions of the axisymmetric moist Hadley circulation in a model with two vertical modes

Abstract: A simplified model of the moist axisymmetric Hadley circulation is examined in the asymptotic limit in which surface drag is strong and the meridional wind is weak compared to the zonal wind. Our model consists of the quasi-equilibrium tropical circulation model (QTCM) equations on an axisymmetric aquaplanet equatorial beta-plane. This model includes two vertical momentum modes, one baroclinic and one barotropic. Prior studies use either continuous stratification, or a shallow water system best viewed as representing the upper troposphere. The analysis here focuses on the interaction of the baroclinic and barotropic modes, and the way in which this interaction allows the constraints on the circulation known from the fully stratified case to be satisfied in an approximate way. The dry equations, with temperature forced by Newtonian relaxation towards a prescribed radiative equilibrium, are solved first. To leading order, the resulting circulation has a zonal wind profile corresponding to uniform angular momentum at a level near the tropopause, and zero zonal surface wind, owing to the cancelation of the barotropic and baroclinic modes there. The weak surface winds are calculated from the first-order corrections. The broad features of these solutions are similar to those obtained in previous studies of the dry Hadley circulation. The moist equations are solved next, with a fixed sea surface temperature at the lower boundary and simple parameterizations of surface fluxes, deep convection, and radiative transfer. The solutions yield the structure of the barotropic and baroclinic winds, as well as the temperature and moisture fields. In addition, we derive expressions for the width and strength of the equatorial precipitating region (ITCZ) and the width of the entire Hadley circulation. The ITCZ width is on the order of a few degrees in the absence of any horizontal diffusion and is relatively insensitive to parameter variations.

Vector potential vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

Abstract: The core dynamics of the Stokes eigenmodes subjected to satisfy no-slip boundary conditions on any 2D closed contour, or 3D surface, is shown to be characterized by a linear differential functional relationship between the vector potential and vorticity, $d\vomega=-\lambda\,d\vpsi$, $\lambda$ being the associated eigenvalue.

Transformations for temperature flux in multiscale models of the tropics

Abstract: How much of the observed planetary-scale heating in the tropics is due to eddy flux convergence? A mathematical framework to address this important practical issue is developed here. We describe a pair of velocity transformations that remove components of the upscale temperature flux in the multiscale intraseasonal, planetary, equatorial synoptic-scale dynamics (IPESD) framework derived by Majda and Klein [J. Atmos. Sci. 60: 393–408, (2003)]. Using examples from the models of the Madden-Julian Oscillation of Biello and Majda [Proc. Natl. Acad. Sci. 101: 4736–4741, (2004); J. Atmos. Sci. 62: 1694–1721, (2005); Dyn. Oceans Atmos., in press] we demonstrate that the transformation for the meridional temperature flux convergence is possible with any restrictions on the heating profile, we show under which conditions the transformation for the vertical temperature flux convergence exists and, further, that the meridional transformation leads to a reinterpretation of lower troposphere Ekman dissipation as active heating plus zonal momentum drag. The meridional temperature flux transformation and induced meridional circulation is a new, tropical wave example of the transformed Eulerian mean theory in the case of strong vertical stratification of potential temperature. The asymptotic ordering of the flows means that the removal of the meridional temperature flux convergence has implications for how planetary-scale heating rates are inferred from velocity convergence measurements.

Nonlinear shallow-water solutions using the weak temperature gradient approximation

Abstract: A form of the weak temperature gradient (WTG) approximation, in which the temperature tendency and advection terms are neglected in the temperature equation so that the equation reduces to a diagnostic balance between heating and vertical motion, is applied to a two-dimensional nonlinear shallow-water model with the heating (mass source) parameterized as a Newtonian relaxation on the temperature (layer thickness) towards a prescribed function of latitude and longitude, containing an isolated maximum or minimum, as in the classic linear Gill problem. In this model, temperature variations are retained in the Newtonian heating term, so that it is not a pure WTG system. It contains no free unbalanced modes, but reduces to the Gill model in the steady linear limit, so that steady solutions may be thought of as containing components corresponding to unbalanced modes in the same sense as the latter. The equations are solved numerically and are compared with full shallow-water solutions in which the WTG approximation is not made. Several external parameters are varied, including the strength, location, sign, and horizontal scale of the mass source, the Rayleigh friction coefficient, and the time scale for the relaxation on the mass field. Indices of the Walker and Hadley circulations are examined as functions of these external parameters. Differences between the WTG solutions and those from the full shallow-water system are small over most of the parameter regime studied, which includes time-dependent as well as steady solutions.

Direct Spatial Resonance in the Compressible Boundary Layer on a Rotating-disk

Abstract: This paper focuses on a resonance mechanism that can lead to significant disturbance amplification at conditions which are sub-critical to nonlinear instabilities Particularly, direct spatial resonance instability is investigated, which is present in the basic three-dimensional viscous compressible boundary-layer flow due to a rotating-disk Within this purpose, the linearized system of stability equations is treated numerically making use of a spectral Chebyshev collocation method The analysis provides critical resonant Reynolds numbers above which growth occurs Amplitudes of the response of the degeneracies decaying rapidly due to their high damping rates are shown to exist for small enough Reynolds numbers while the flow is still in the laminar state If the flow is restricted to the incompressible case, the results of Turkyilmazoglu and Gajjar (in Sadhana Acad P Engs 25:601–617, 2000) are completely reproduced The influences of compressibility are then explored by means of varying the Mach and Prandtl numbers in the cases of heating/cooling the wall as well as the isothermal wall In general, compressibility effects are found strongly in favor of stabilizing as the Mach number increases, while a strong destabilization is observed by lowering the critical values of Reynolds numbers in the cases of wall heating and insulation The modal interaction and coalescence of the eigenmodes calculated here create local algebraic growth by rapid development of relatively large amplitudes which might then provide the onset of nonlinear effects followed by transition

Complementary modes of impulse generation for flow past a three-dimensional obstacle

Abstract: A Lagrangian model for viscous incompressible flow past a stationary solid obstacle is developed in terms of the creation over successive instants of time of fluid impulse at the obstacle’s bounding surface. Upon creation this compactly supported vector field actively evolves into the flow interior according to its equation of motion. At the wall, distinct creation modes can be identified by specifying two exterior Poisson problems, distinguished from each other by complementary gradient boundary conditions. We demonstrate how in each case the relevant Poisson problem is to be solved numerically. Impulse generation is modeled in Lagrangian terms for each mode in turn for the case of flow past a sphere. (In the spherical geometry we exploit image theory to simplify some aspects of the computation.) We argue that the concurrent existence of these two distinct modes provides an insight into the complex phenomenology of turbulent wake formation.