Kinetic Energy Budget Analysis Research Articles

Turbulent kinetic dissipation analysis for residual-based large eddy simulation of incompressible turbulent flow by variational multiscale method

The underlying physical mechanism of the residual-based large eddy simulation (LES) based on the variational multiscale (VMS) method is clarified. Resolved large-scale energy transportation equation is mathematically derived for turbulent kinetic energy budget analysis. Firstly, statistical results of benchmark turbulent channel flow at Reτ=180 obtained using a coarse mesh are compared with the results obtained by the classical LES with the Smagorinsky and dynamic subgrid stress (SGS) model. The present LES shows an advantage in predicting the statistical results of the incompressible turbulent flows. Secondly, the contributions of the unresolved small-scale presentation terms (Term I-IV in Eq. (10)) to the turbulent kinetic dissipation are analysed for the VMS method. The results show that the turbulent kinetic dissipation provided by the numerical diffusion in the VMS method is smaller in the inner layer, larger in the outer layer of the channel flow than those by the Smagorinsky and dynamic SGS model. The turbulent kinetic dissipation in the VMS method is mainly given by the numerical diffusion provided by one of the “cross-stress” terms (Term I, same as the stabilization term in the SUPG method) and LSIC term (Term IV). The other one of the “cross-stress” terms (Term II) gives rise to the positive turbulent kinetic energy budget, and does not dissipate the turbulent kinetic energy. The so-called “Reynolds stress” term (Term III) dissipates the turbulent energy but provides a very small numerical diffusion. Finally, on the basis of the turbulent kinetic energy dissipation analysis, a new residual-based stabilized finite element formulation is proposed by modifying the large-scale equation in the VMS method. Numerical experiments of 2D lid-driven cavity flow and 3D incompressible turbulent channel flow are tested to validate the proposed formulation. It is shown that all the stabilization terms in the proposed formulation produce additional numerical diffusions and physically increase the total turbulent kinetic dissipation. Consequently, an apparent improvement in both the first-order and second-order statistical quantities are pursued by the new stabilized finite element formulation.

New findings in vorticity dynamics of turbulent buoyant plumes

The vorticity dynamics of turbulent, buoyant, and pure plumes released into the quiescent ambience has been investigated using large-eddy simulations with plume Reynolds number in the range of 6 × 107 to 108. The plume is generated by a circular heat source and sustained by buoyancy forcings generated by heating. As the starting plume rises vertically, it expands radially, entraining ambient fluid into the plume and two distinct stages of evolution are evident. During stage one (initial stage), in the near-source region the plume is accelerating characterized by developing turbulence. During the next stage (mixing stage), the plume is significantly altered by turbulence resulting in significant entrainment and expansion in the radial direction. As turbulent intensities decay during the mixing stage, the enstrophy decays in an exponential manner with height with an exponent of −7/4. The turbulent kinetic energy budget analysis reveals—baroclinic torque, stretching, and compression—as the three dominant mechanisms for the plume growth. The probability distribution function (PDF) of vorticity shows that vorticity is mainly aligned along the transverse direction near the source and slowly reaches to a quasi-isotropy state downstream. Turbulence spectra demonstrate the presence of a buoyancy-regime with a −3 spectral slope. The PDF of vorticity further shows extreme dominance of the strain rate rather than rotation within the plume. Consistent with studies in the literature, the vorticity fluctuations align with the intermediate eigen-vector of strain-rate tensor.

Characterization of the shear layer in separated shock/turbulent boundary layer interactions

Abstract

The Tropical Cyclone as a Divergent Source in a Background Flow

AbstractThe interactions between the outflow of a tropical cyclone (TC) and its background flow are explored using a hierarchy of models of varying complexity. Previous studies have established that, for a select class of TCs that undergo rapid intensification in moderate values of vertical wind shear, the upper-level outflow of the TC can block and reroute the environmental winds, thus reducing the shear and permitting the TC to align and subsequently to intensify. We identify in satellite imagery and reanalysis datasets the presence of tilt nutations and evidence of upwind blocking by the divergent wind field, which are critical components of atypical rapid intensification. We then demonstrate how an analytical expression and a shallow water model can be used to explain some of the structure of upper-level outflow. The analytical expression shows that the dynamic high inside the outflow front is a superposition of two pressure anomalies caused by the outflow’s deceleration by the environment and by the environment’s deceleration by the outflow. The shallow water model illustrates that the blocking is almost entirely dependent upon the divergent component of the wind. Then, using a divergent kinetic energy budget analysis, we demonstrate that, in a full-physics TC, upper-level divergent flow generation occurs in two phases: pressure driven and then momentum driven. The change happens when the tilt precession reaches left of shear. When this change occurs, the outflow blocking extends upshear. We discuss these results with regard to prior severe weather studies.

Submesoscale Surface Tidal, Vortical, and Residual Circulations in a Semienclosed Bay

AbstractWe scrutinize three different components of submesoscale surface tidal, vortical, and residual circulations in a semienclosed bay using (1) observations of high‐frequency radar‐derived surface current maps and (2) numerical model simulations run under realistic vertical stratification and boundary conditions at O(1)‐km spatial and 1‐hourly temporal resolutions over a 2‐year period (2013‐2014). The tidal circulation is characterized by (1) the tidal ellipses and spectral contents having more baroclinic motions at the M2 and S2 frequencies and more barotropic motions at the K1 frequency and (2) the temporal and spatial variability in the tidal fronts appearing as tidal ellipses with clustered shapes and opposite rotational directions. The regional vortical circulation is presented with the submesoscale eddies at diameters ranging from 3 to 12 km and normalized vorticity magnitudes of 0.2 to 2 for both clockwise and counterclockwise rotations. Based on the spatial statistics of the identified submesoscale eddies and eddy kinetic energy budget analysis, the submesoscale eddies are primarily generated by the detachment of shoreline‐following tidal currents at the coastal boundaries, persist for less than 1.5 days, and are dissipated dominantly via vertical buoyancy fluxes associated with bottom bathymetric interactions of the tidal currents. The residual circulation is clearly shown with the nontidal geostrophic currents associated with the pressure gradients generated by wind‐driven Ekman transports against the coast and the ageostrophic low‐frequency currents.

Kinetic energy spectra and flux in turbulent phase-separating symmetric binary-fluid mixtures

We conduct direct numerical simulations (DNS) of the Cahn–Hilliard–Navier–Stokes (CHNS) equations to investigate the statistical properties of a turbulent phase-separating symmetric binary-fluid mixture. Turbulence causes an arrest of the phase separation which leads to the formation of a statistically steady emulsion. We characterise turbulent velocity fluctuations in an emulsion for different values of the Reynolds number and the Weber number. Our scale-by-scale kinetic energy budget analysis shows that the interfacial terms in the CHNS equations provide an alternative route for the kinetic energy transfer. By studying the probability distribution function (p.d.f.) of the energy dissipation rate, the vorticity magnitude and the joint-p.d.f. of the velocity-gradient invariants we show that the statistics of the turbulent fluctuations do not change with the Weber number.

The Universality of the Normalized Vertical Velocity Variance in Contrast to the Horizontal Velocity Variance in the Convective Boundary Layer

AbstractThe vertical turbulent velocity variance normalized by the convective velocity squared as a function of the boundary layer depth–normalized height [i.e., ] in the convective boundary layer (CBL) over a homogeneous surface exhibits a near-universal profile, as demonstrated by field observations, laboratory experiments, and numerical simulations. The profile holds over a wide CBL stability range set by the friction velocity, CBL depth, and surface heating. In contrast, the normalized horizontal turbulent velocity variance increases monotonically with decreasing stability. This study investigates the independence of the profile to changes in CBL stability, or more precisely, wind shear. Large-eddy simulations of several convective and neutral cases are performed by varying surface heating and geostrophic winds. Analysis of the turbulent kinetic energy budgets reveals that the conversion term between and depends almost entirely on buoyancy. This explains why does not vary with shear, which is a source to only. Further analysis through rotational and divergent decomposition suggests that the near-universal profile of is fundamentally related to the dynamics and interactions of local and nonlocal CBL turbulence. Specifically, the preferential interactions between local wavenumbers and the downscale energy cascade of CBL turbulence offer plausible explanations to the universal profile of .

Kinetic energy budget and moisture flux convergence analysis during interaction between two cyclonic systems: Case study

An analysis of the kinetic energy budget during a case of interaction between middle latitude and extratropical cyclones has been made in this work. Horizontal flux convergence constitutes a major energy sink. Generation of kinetic energy via cross-contour flow is a persistent source throughout the growth and decay periods. Dissipation of kinetic energy from subgrid to grid scales is an important source during the pre-storm period; it acts as a sink during the growth and decay periods. The major contribution to kinetic energy comes from a persistent upper tropospheric jet stream activity throughout the period of the cyclone development. The characteristics of moisture-flux components (divergent and rotational) along with precipitable water content for different tropospheric layers throughout the life cycle of our cyclone are also studied in this work. It is found that most of required humidity for our cyclone are initiated from Arabian Sea and then to some extent are reinforced over Gulf of Aden and east of central Africa and then by passing over Red sea enter to the south and south east of Mediterranean Sea. The rotational component of the moisture transport brings moisture from two regions; the first which is considered the main region is the Indian Ocean, Arabian Sea, Gulf of Aden and north east of Sudan. The second source region is the Atlantic and Mediterranean Sea. In the middle troposphere, the primary moisture source is found over central Africa, which in turn is traced to the Atlantic Ocean, the Indian Ocean, and the Arabian Sea. The upper-level moisture fluxes are weak and play a minor role over the area of interaction between two cyclones.

Assessment of Computational Fluid Dynamic for Surface Combatant 5415 at Straight Ahead and Static Drift β = 20 deg

Collaboration is described on assessment of computational fluid dynamics (CFD) predictions for surface combatant model 5415 at static drift β = 0 deg and 20 deg using recent tomographic particle image velocimetry (TPIV) experiments. Assessment includes N-version verification and validation to determine the confidence intervals for CFD solutions/codes, and vortex onset, progression, instability, and turbulent kinetic energy (TKE) budget analysis. The increase in β shows the following trends. Forces and moment increase quadratically/cubically, and become unsteady due to shear layer, Karman and flapping instabilities on the bow. Wave elevation becomes asymmetric; its amplitude increases, but the total wave elevation angle remains same. The vortex strength and TKE increase by about two orders of magnitude, and for large β, the primary vortices exhibit helical mode instability similar to those for delta wings. Forces and moment for both β and wave elevation for β = 0 deg are compared within 4% of the data, and are validated at 7% interval. Wave elevation for β = 20 deg, and vortex core location and velocities for both β are compared within 9% of the data, and are validated at 12% interval. The vortex strength and TKE predictions show large 70% errors and equally large scatter and are not validated. Thus, both errors and scatter need reduction. TKE budgets show transport of turbulence into the separation bubble similar to canonical cases, but pressure transport is dominant for ship flows. Improved CFD predictions require better grids and/or turbulence models. Investigations of solution-adaptive mesh refinement for better grid design and hybrid Reynolds-averaged Navier-Stokes/large eddy simulation models for improved turbulent flow predictions are highest priority.

Lack of Westerly Wind Bursts in Unmaterialized El Niño Years

The lack of westerly wind bursts (WWBs) when atmospheric intraseasonal variability (ISV) events occur from boreal spring to autumn is investigated by comparing two types of El Niño years with unmaterialized El Niño (UEN) years. Although high ocean heat content buildup and several ISV events propagating eastward are observed in all three types of years, few WWBs accompany these in the UEN years. The eddy kinetic energy budget analysis based on ISV shows that mean westerly winds in the lower troposphere facilitate the development of eddy disturbances, including WWBs, through convergence and meridional shear of zonal winds. In the UEN years, these westerly winds are retracted westward and do not reach the equatorial central Pacific mainly as a result of interannual components. In addition, positive sea surface temperature anomalies in the western Pacific, which are conducive to active convection, spread widely in a meridional direction centered on 15°N. Both westward-retracted mean westerlies and off-equatorial warming enhance off-equatorial eddies, which result in a reduction in equatorial eddies such as WWBs. The characteristics of the UEN years are significantly different from those observed during the eastern Pacific El Niño (EP-EN) years, which are characterized by anomalous cooling (warming) and suppressed (enhanced) convective eddies in the off-equatorial (equatorial) western Pacific. The central Pacific El Niño years show mixed features during both EP-EN and UEN years. Different background states not only in the equatorial region but also in the off-equatorial region can be a reason for the lack of WWBs in the UEN years.

An Eulerian two-phase model for steady sheet flow using large-eddy simulation methodology

Abstract A three-dimensional Eulerian two-phase flow model for sediment transport in sheet flow conditions is presented. To resolve turbulence and turbulence-sediment interactions, the large-eddy simulation approach is adopted. Specifically, a dynamic Smagorinsky closure is used for the subgrid fluid and sediment stresses, while the subgrid contribution to the drag force is included using a drift velocity model with a similar dynamic procedure. The contribution of sediment stresses due to intergranular interactions is modeled by the kinetic theory of granular flow at low to intermediate sediment concentration, while at high sediment concentration of enduring contact, a phenomenological closure for particle pressure and frictional viscosity is used. The model is validated with a comprehensive high-resolution dataset of unidirectional steady sheet flow (Revil-Baudard et al., 2015, Journal of Fluid Mechanics, 767, 1–30). At a particle Stokes number of about 10, simulation results indicate a reduced von Karman coefficient of κ ≈ 0.215 obtained from the fluid velocity profile. A fluid turbulence kinetic energy budget analysis further indicates that the drag-induced turbulence dissipation rate is significant in the sheet flow layer, while in the dilute transport layer, the pressure work plays a similar role as the buoyancy dissipation, which is typically used in the single-phase stratified flow formulation. The present model also reproduces the sheet layer thickness and mobile bed roughness similar to measured data. However, the resulting mobile bed roughness is more than two times larger than that predicted by the empirical formulae. Further analysis suggests that through intermittent turbulent motions near the bed, the resolved sediment Reynolds stress plays a major role in the enhancement of mobile bed roughness. Our analysis on near-bed intermittency also suggests that the turbulent ejection motions are highly correlated with the upward sediment suspension flux, while the turbulent sweep events are mostly associated with the downward sediment deposition flux.

Effect of polymer additives on heat transport and large-scale circulation in turbulent Rayleigh-Bénard convection.

The present paper presents direct numerical simulations of Rayleigh-Bénard convection (RBC) in an enclosed cell filled with the polymer solution in order to investigate the viscoelastic effect on the characteristics of heat transport and large-scale circulation (LSC) of RBC. To overcome the difficulties in numerically solving a high Weissenberg number (Wi) problem of viscoelastic fluid flow with strong elastic effect, the log-conformation reformulation method was implemented. Numerical results showed that the addition of polymers reduced the heat flux and the amount of heat transfer reduction (HTR) behaves nonmonotonically, which firstly increases but then decreases with Wi. The maximum HTR reaches around 8.7% at the critical Wi. The nonmonotonic behavior of HTR as a function of Wi was then corroborated with the modifications of the period of LSC and turbulent energy as well as viscous boundary layer thickness. Finally, a standard turbulent kinetic energy (TKE) budget analysis was done for the whole domain, the boundary layer region, and the bulk region. It showed that the role change of elastic stress contributions to TKE is mainly responsible for this nonmonotonic behavior of HTR.

Numerical studies of shock wave interactions with a supersonic turbulent boundary layer in compression corner:Turning angle effects

Abstract Direct numerical simulations (DNS) were performed to investigate the interactions of a Mach 2.9 turbulent boundary layer with shock waves of varying strengths in compression corner. The supersonic turbulent boundary layer was triggered by wall blowing-and-suction perturbations. The shock waves were produced by two-dimensional compression corners of 8, 14, 20 and 24°. Compared with previous DNS results and experimental data, the numerical calculations were validated. The effects of shock wave on the boundary layer are studied by both flow visualizations and statistical analysis, and the results show that the intensity of fluctuations is amplified greatly by the shock wave. With the increasing of turning angle, three-dimensionality of separation bubble is significantly enhanced. Based on the statistics and power spectrum of the wall pressure signals, the effect of turning angle on the unsteadiness of shock motion is also studied, and the results show that the shock motions are quite different in the small and the large turning angle cases. The motion in the 8° and 14° cases is characterized by high-frequency and small-amplitude, but the low-frequency and large-scale streamwise oscillation is the main feature in the 20° and 24° cases. The effect of turning angle on the turbulence state is analyzed by using the anisotropy of Reynolds stress tensor. The coherent vortex structures are also studied qualitatively. The results indicate that the cane-like streamwise vortexes in the near-wall region are the dominant structure for the small angle cases, while the hairpin vortexes and packets in the outer layer play the leading role in the large angle cases. According to the quantitative analysis of turbulent kinetic energy budgets in the separation region, the effect of turning angle on the transport mechanism is studied. It is found that the influence of shear layer above separation bubble on the mechanism is significant.

Global Stability Analyses Unraveling Roughness-induced Transition Mechanisms

The linear global instability and resulting transition to turbulence induced by a cylindrical roughness element of height h and diameter d = 3h immersed within an incompressible boundary layer flow along a flat plate is investigated using the joint application of direct numerical simulations and three-dimensional stability analyses. The configuration investigated is the same as the one in- vestigated experimentally by Fransson et al.10. Base flow computations show that the roughness element induces a wake composed of a central low-speed region surrounded by a three-dimensional shear layer and a pair of low- and high-speed streaks on each side. Results from the global stability analyses highlight the unstable nature of the central low-speed region and its crucial importance in the laminar-turbulent transition process. For the set of parameters considered, it is able to sustain a varicose global instability for which the predicted critical Reynolds number is only 6% larger than the one reported in Ref. 10. A kinetic energy budget and wavemaker analysis revealed that this mode finds its root in the reversed flow region right downstream the roughness element and extracts most of its energy from the central low-speed region and streaks further downstream. Direct numerical simulations of the flow past this roughness element puts in the limelight the ability for this linear instability to give birth to hairpin vortices and thus trigger transition to turbulence.

Effect of subgrid-scale mixing on the evolution of forced submesoscale instabilities

We study the effect of subgrid-scale (SGS) mixing on the evolution of a density front initially in thermal-wind balance with a meridional density gradient and forced by downfront surface winds. The horizontal size of the model domain (O(100km)) is large enough to contain mesoscale eddies while the horizontal grid resolution (500m) is fine enough to resolve submesoscale eddies. The twin goals of this study are: (i) to determine what is a realistic level of SGS dissipation; and (ii) to explore the sensitivity of the resolved-scale dynamics to the SGS dissipation. To this end, we effect different levels of SGS dissipation using two SGS models: (i) constant lateral SGS viscosities (1m2s−1 and 5m2s−1) and an analytically prescribed vertical SGS viscosity; and (ii) an existing anisotropic Smagorinsky model (ASM) developed for anisotropic grids with large aspect ratios between the horizontal and the vertical directions. An analysis of the eddy kinetic energy (EKE) budgets shows the surface stress boundary condition constrains all simulations to yield realistic values of SGS dissipation in a near-surface layer that is shear-driven and similar to the traditional Monin–Obukhov layer. Deeper down within the mixed layer, the EKE budget is buoyancy-driven with a more complicated balance that varies considerably among the different simulations. The simulations with constant Kx predict the buoyant generation of EKE is balanced almost solely by pressure transport with negligible local destruction, which gives rise to waves near the front. Recent observations near fronts show enhanced levels of irreversible destruction. The simulations with the ASM predict EKE budgets where both local destruction—through SGS dissipation—and pressure transport are part of the EKE balance.The results obtained using the constant-Kx simulations suggest both horizontal and vertical SGS parameterizations have important effects on the resolved-scale dynamics. The simulations with Kx=5m2s−1 yield the most unrealistic results partly because the lateral viscosity is high enough to directly influence the instability scale. Yet the observed differences among the constant-Kx simulations are sometimes subtle and cannot be explained trivially by comparing Kx alone. For fixed Kx, we find simulations can exhibit higher spectral levels and stronger cascades (forward and inverse) upon increasing the vertical SGS viscosity. This suggests the sensitivity of submesoscale-resolving simulations to the vertical SGS parameterization needs to be better explored.

Warm Conveyor Belts in Idealized Moist Baroclinic Wave Simulations

AbstractThis idealized modeling study of moist baroclinic waves addresses the formation of moist ascending airstreams, so-called warm conveyor belts (WCBs), their characteristics, and their significance for the downstream flow evolution. Baroclinic wave simulations are performed on the f plane, growing from a finite-amplitude upper-level potential vorticity (PV) perturbation on a zonally uniform jet stream. This nonmodal approach allows for dispersive upstream and downstream development and for studying WCBs in the primary cyclone and the downstream cyclone. A saturation adjustment scheme is used as the only difference between the dry and moist simulations, which are systematically compared using a cyclone-tracking algorithm, with an eddy kinetic energy budget analysis, and from a PV perspective. Using trajectories and a selection criterion of maximum ascent, forward- and rearward-sloping WCBs in the moist simulation are identified. No WCB is identified in the dry simulation. Forward-sloping WCBs originate in the warm sector, move into the frontal fracture region, and ascend over the bent-back front, where maximum latent heating occurs in this simulation. The outflow of these WCBs is located at altitudes with prevailing zonal winds; they hence flow anticyclonically (“forward”) into the downstream ridge. In case of a slightly weaker ascent, WCBs curve cyclonically (“rearward”) above the cyclone center. A detailed analysis of the PV evolution along the WCBs reveals PV production in the lower troposphere and destruction in the upper troposphere. Consequently, WCBs transport low-PV air into their outflow region, which contributes to the formation of distinct negative PV anomalies. They, in turn, affect the downstream flow and enhance downstream cyclogenesis.

Dynamical influence of gravity waves generated by the Vestfjella Mountains in Antarctica: radar observations, fine-scale modelling and kinetic energy budget analysis

ABstractGravity waves generated by the Vestfjella Mountains (in western Droning Maud Land, Antarctica, southwest of the Finnish/Swedish Aboa/Wasa station) have been observed with the Moveable atmospheric radar for Antarctica (MARA) during the SWEDish Antarctic Research Programme (SWEDARP) in December 2007/January 2008. These radar observations are compared with a 2-month Weather Research Forecast (WRF) model experiment operated at 2 km horizontal resolution. A control simulation without orography is also operated in order to separate unambiguously the contribution of the mountain waves on the simulated atmospheric flow. This contribution is then quantified with a kinetic energy budget analysis computed in the two simulations. The results of this study confirm that mountain waves reaching lower-stratospheric heights break through convective overturning and generate inertia gravity waves with a smaller vertical wavelength, in association with a brief depletion of kinetic energy through frictional dissipation and negative vertical advection. The kinetic energy budget also shows that gravity waves have a strong influence on the other terms of the budget, i.e. horizontal advection and horizontal work of pressure forces, so evaluating the influence of gravity waves on the mean-flow with the vertical advection term alone is not sufficient, at least in this case. We finally obtain that gravity waves generated by the Vestfjella Mountains reaching lower stratospheric heights generally deplete (create) kinetic energy in the lower troposphere (upper troposphere–lower stratosphere), in contradiction with the usual decelerating effect attributed to gravity waves on the zonal circulation in the upper troposphere–lower stratosphere.

Stability of fluids with shear-dependent viscosity in the lid-driven cavity

The classical problem of the lid-driven cavity extended infinitely in the spanwise direction is considered for non-Newtonian shear-thinning and shear-thickening fluids, where the viscosity is modeled by the Carreau model. Linear stability is used to determine the critical Reynolds number at which the two-dimensional base-flow becomes unstable to three-dimensional spanwise-periodic disturbances. We consider a square cavity, characterized by steady unstable modes, and a shallow cavity of aspect ratio 0.25, where oscillating modes are the first to become unstable for Newtonian fluids. In both cases, the critical Reynolds number first decreases with decreasing power-index n (from shear-thickening to shear-thinning fluids) and then increase again for highly pseudoplastic fluids. In the latter case, this is explained by the thinner boundary layers at the cavity walls and less intense vorticity inside the domain. Interestingly, oscillating modes are found at critical conditions for shear-thickening fluids in a square cavity while the shallow cavity supports a new instability of lower frequency for large enough shear-thinning. Analysis of kinetic energy budgets and structural sensitivity are employed to investigate the physical mechanisms behind the instability.

Kinetic Energy Budget Analysis on the Development of a Meso-β-Scale Vortex Causing Heavy Rainfall, Observed over Aomori Prefecture in Northern Japan on 11 November 2007

Kinetic Energy Budget Analysis on the Development of a Meso-β-Scale Vortex Causing Heavy Rainfall, Observed over Aomori Prefecture in Northern Japan on 11 November 2007

Kinetic energy budget analysis of spiraling columnar critical convection in a rapidly rotating spherical shell

The kinetic energy budget analysis of spiraling columnar critical convection emerging in a rapidly rotating spherical shell is performed. In the cylindrical-radially elongated spiraling convection mode appearing as a critical mode at small Prandtl numbers, the kinetic energy generated in the inner region of the spherical shell is transported in a cylindrically radial manner and is dissipated near the outer boundary around the equator. However, when the Prandtl number is increased, the dynamical energy flux turns in the axial direction rather than in the cylindrically radial direction. The kinetic energy generated inside the shell is transported in the direction of the rotating axis and is dissipated near the outer boundary at the same cylindrically radial location. The existence of the axial component of the dynamical energy flux is attributed to the ageostrophic flows in the columnar vortices induced by viscous damping and buoyancy force. In spite of the existence of the axial component of dynamical energy flux, the obtained geometry of the axially integrated kinetic energy budget is consistent with the results using a two-dimensional rotating annulus model. Therefore, the interpretation of spiraling structure with the radial propagation properties of topographic Rossby waves is applicable to the three-dimensional spiraling columnar convection emerging in a rotating spherical shell. Flow patterns calculated from the dispersion relation of two-dimensional topographic Rossby waves in a rotating spherical shell well explain the structure of the three-dimensional spiraling columnar convection.