Mean Momentum Equations Research Articles

On the Lundgren hierarchy of helically symmetric turbulence

Abstract This paper analyzes the reduction of the infinite Lundgren-Monin-Novikov (LMN) hierarchy of probability density functions (PDFs) in the statistical theory of helically symmetric turbulence. Lundgren's hierarchy is considered a complete model, i.e. fully describes the joint multi-point statistic of turbulence though at the expense of dealing with an infinite set of integro-differential equations. The LMN hierarchy and its respective side-conditions are transformed to helical coordinates and thus are dimesionally reduced. In the course of development, a number of key questions were solved, namely in particular the transformation of PDFs and sample space velocities into orthonormal coordinate systems. In a validity check it is shown, that the mean momentum equations derived from the helical LMN hierarchy via statistical moment integration are identical to the mean momentum equations derived by direct ensemble averaging the Navier-Stokes equation, in helically symmetric form. Finally, we derive the equation for the characteristic function equivalent to the PDF equation in a helically symmetric frame, which allows to generate arbitrary n^th-order statistical moments by simple differentiation.

A momentum budget study of the semi‐annual oscillation in the Whole Atmosphere Community Climate Model

AbstractThe representation of the semi‐annual oscillation (SAO) in climate models shows a common easterly bias of several tens of metres per second compared to observations. These biases could be due to deficiencies in eastward tropical wave forcing, the position or strength of the climatological summertime jet or the strength/timing of the Brewer–Dobson circulation. This motivates further analysis of the momentum budget of the upper stratosphere within models and a more detailed comparison with reanalyses to determine the origin of the bias. In this study, the transformed Eulerian mean momentum equation is used to evaluate the different forcing terms that contribute to the SAO in the MERRA2 reanalysis dataset. This is then compared with the equivalent analysis using data from a climate simulation of the Whole Atmosphere Community Climate Model (WACCM). The comparison shows that WACCM underestimates eastward forcing by both resolved and parameterised waves at equatorial latitudes when compared with MERRA2 and also has a weaker tropical upwelling above 1 hPa.

Local Similarity Theory as the Invariant Solution of the Governing Equations

The present paper shows that local similarity theories, proposed for the strongly-stratified boundary layers, can be derived as invariant solutions defined under the Lie-group theory. A system truncated to the mean momentum and buoyancy equations is considered for this purpose. The study further suggests how similarity functions for the mean profiles are determined from the vertical fluxes, with a potential dependence on a measure of the anisotropy of the system. A time scale that is likely to characterize the transiency of a system is also identified as a non-dimensionalization factor.

Numerical investigation of mixed-phase turbulence induced by a plunging jet

In nature and engineering applications, water jet plunging acts as a key process causing interface breaking and generating mixed-phase turbulence. In this paper, high-resolution numerical simulations of the plunging of a water jet into a quiescent pool were performed to investigate the statistical properties of mixed-phase turbulence, with a special focus on the closure problem of the Reynolds-averaged equation. We conducted phase-resolved simulations, with the air–water interface captured using a coupled level-set and volume-of-fluid method. Various cases were performed to analyse the effects of the Froude number and Reynolds number. The simulation results showed that the turbulence statistics are insensitive to the Reynolds number under investigation, while the Froude number influences the flow properties significantly. To investigate the closure problem of the mean momentum equation, the turbulent kinetic energy (TKE) and turbulent mass flux (TMF) and their transport equations were analysed further. It was discovered that the balance relationship of the TKE budget terms remained similar to many single-phase turbulent flows. The TMF is an additional unclosed term in mixed-phase turbulence over the single-phase turbulence. Our simulation results showed that the production term in its transport equation was highly correlated to TKE. Based on this finding, a closure model for the production term of TMF was further proposed.

Velocity and temperature scalings leading to compressible laws of the wall

We exploit the similarity between the mean momentum equation and the mean energy equation and derive transformations for mean temperature profiles in compressible wall-bounded flows. In contrast to prior studies that rely on the strong Reynolds analogy and the presumed similarity between the instantaneous and mean velocity and temperature signals, the discussion in this paper involves the Farve-averaged equations only. We establish that the compressible momentum and energy equations can be made identical to their incompressible counterparts under appropriate normalizations and coordinate transformations. Two types of transformations are explored for illustration purposes: Van Driest (VD)-type transformations and semi-local-type or Trettel–Larsson (TL)-type transformations. In our derivations, it becomes clear that VD-type velocity and temperature transformations hold exclusively within the logarithmic layer. On the other hand, TL-type transformations extend their applicability to incorporate wall-damping effects, at least in principle. Each type of transformation serves its distinct purpose and has its applicable range. However, it is noteworthy that while VD-type transformations can be assessed using measurements obtained from laboratory experiments, TL-type transformations necessitate viscosity and density information typically accessible only through numerical simulations. Finally, we justify the omission of the turbulent kinetic energy transfer term, a term that is unclosed, in the energy equation. This omission leads to closed-form temperature transformations that are valid for both adiabatic and isothermal walls.

New formulations for the mean wall-normal velocity and Reynolds shear stress in a turbulent boundary layer under zero pressure gradient

In this study, a novel analytical solution is derived rigorously for the mean wall-normal velocity in a turbulent boundary layer (TBL) under zero pressure gradient (ZPG), without relying on any a priori assumptions regarding the mean streamwise velocity. By neglecting the higher-order terms in the exact solution, an approximate formulation for the mean wall-normal velocity is obtained. The accuracy of this approximation is then validated through a comparison with data from direct numerical simulations (DNS). Drawing upon the distinct behaviours of force balance in the inner and outer regions of the ZPG TBL, simplified approximate mean momentum equations are derived by decomposing the Reynolds shear stress into inner and outer parts. The inner and outer Reynolds shear stresses are then obtained through integration of the corresponding approximate mean momentum equations. The outer Reynolds shear stress is revealed to exhibit a strong connection to the mean wall-normal velocity. Specifically, it is observed that $R_{uv-{out}}/u^2_\tau \approx 1 - UV/(U_e V_e)$ , where $u_\tau$ represents the friction velocity, and $U_e$ and $V_e$ denote the mean streamwise and wall-normal velocities at the boundary layer edge, respectively. Finally, an approximate formulation for the Reynolds shear stress across the entire TBL is developed by synthesizing the inner and outer parts. Extensive validation against both DNS and experimental data, spanning a wide range of Reynolds numbers, demonstrates the excellent agreement achieved.

Outer scaling of the mean momentum equation for turbulent boundary layers under adverse pressure gradient

A new scaling of the mean momentum equation is developed for the outer region of turbulent boundary layers (TBLs) under adverse pressure gradient (APG). The maximum Reynolds shear stress location, denoted as $y_{m}$ , is employed to determine the proper scales for the outer region of an APG TBL. An outer length scale is proposed as $\delta _e - y_{m}$ , where $\delta _e$ is the boundary layer thickness. An outer velocity scale for the mean streamwise velocity deficit is proposed as $U_e - U_{m}$ , where $U_e$ and $U_m$ are the mean streamwise velocities at the boundary layer edge and $y_{m}$ , respectively. An outer velocity scale for the mean wall-normal velocity deficit is proposed as $V_e - V_{m}$ , where $V_e$ and $V_{m}$ are the wall-normal velocities at $\delta _e$ and $y_{m}$ , respectively. The maximum Reynolds shear stress is found to scale as $(\delta _e - y_{m}) U_e \,{\rm d}U_e/{{\rm d}x}$ . The new outer scaling collapses well the experimental and numerical data on APG TBLs over a wide range of Reynolds numbers and strengths of pressure gradient. Approximations of the new scaling are developed for TBLs under strong APG and at high Reynolds numbers. The relationships between the new scales and previously proposed scales are discussed.

Development of explicit algebraic models of Reynolds stresses for flows in channels using gene expression programming

To build a data-driven approximation for the Reynolds-stress anisotropy (RSA), the symbolic regression method of gene expression programming (GEP) is applied. Two tensor-basis terms from the algebraic expansion for the RSA tensor are used in the GEP algorithm. A new RANS-GEP model is tested in several flows in channels without/with bumps at different physical and geometrical parameters, where DNS data of high fidelity involved as a target for RSA are available. The results of RANS-DNS runs are also obtained where the RSA values in the mean momentum equation are taken directly from DNS to show ability to improve the model performance versus the conventional linear eddy viscosity model (LEVM). Next, the training with carefully selected input features is performed to get an explicit non-linear algebraic model for RSA. The results of RANS-GEP study show potentials of the new tool to improve predictions.

Scaling patch analysis of planar turbulent mixing layers

Proper scales for the mean flow and Reynolds shear stress in planar turbulent mixing layers are determined from a scaling patch analysis of the mean continuity and momentum equations. By seeking an admissible scaling of the mean continuity equation, a proper scale for the mean transverse flow is determined as Vref=(dδ/dx)Uref, where dδ/dx is the growth rate of the mixing layer width and Uref=Uh−Ul is the difference between the velocity of the high speed stream Uh and the velocity of the low speed stream Ul. By seeking an admissible scaling for the mean momentum equation, a proper scale for the kinematic Reynolds shear stress is determined as Ruv,ref=UavgVref=[12Audδdx]Uref2, where Au=def(Uh−Ul)/(Uh+Ul) is the normalized velocity difference that emerges naturally in the admissible scaling of the mean momentum equation. Self-similar equations for the scaled mean transverse flow V* and Reynolds shear stress Ruv*=Ruv/Ruv,ref are derived from the mean continuity and mean momentum equations. Approximate equations for V* and Ruv* are developed and found to agree well with experimental data.

Scaling patch analysis of planar turbulent wakes

A scaling patch approach is used to investigate the proper scales in planar turbulent wakes. A proper scale for the mean axial flow is the well-known maximum velocity deficit Uref=U∞−Uctr, where U∞ is the free stream velocity and Uctr is the mean axial velocity at the wake centerline. From an admissible scaling of the mean continuity equation, a proper scale for the mean transverse flow is found as Vref=(dδ/dx)Uref, where dδ/dx is the growth rate of the wake width. From an admissible scaling of the mean momentum equation, a proper scale for the kinematic Reynolds shear stress is found as Ruv,ref=U∞Vref, which is a mixed scale of the free stream velocity and the mean transverse flow scale. Expressions are derived for the scaled mean transverse velocity and Reynolds shear stress in the far field of planar turbulent wakes. Using a Gaussian function for the mean axial velocity deficit, approximate functions for the scaled mean transverse velocity and Reynolds shear stress are developed and found to agree well with experimental and simulation data. This work reveals that the mean transverse flow, despite its small magnitude, plays an important role in the scaling and understanding of the planar turbulent wake.

Layered structure of turbulent plane wall jet

This paper investigates the layered structure of a turbulent plane wall jet at a distance from the nozzle exit. Based on the force balances in the mean momentum equation, the turbulent plane wall jet is divided into three regions: a boundary layer-like region (BLR) adjacent to the wall, a half free jet-like region (HJR) away from the wall, and a plug flow-like region (PFR) in between. In the PFR, the mean streamwise velocity is essentially the maximum velocity, and the simplified mean continuity and mean momentum equations result in a linear variation of the mean wall-normal velocity and Reynolds shear stress. In the HJR, as in a turbulent free jet, a proper scale for the mean wall-normal flow is the mean wall-normal velocity far from the wall and a proper scale for the Reynolds shear stress is the product of the maximum mean streamwise velocity and the velocity scale for the mean wall-normal flow. The BLR region can be divided into four sub-layers, similar to those in a canonical pressure-driven turbulent channel flow or shear-driven turbulent boundary layer flow. Building on the log-law for the mean streamwise velocity in the BLR, a new skin friction law is proposed for a turbulent wall jet. The new prediction agrees well with the correlation of Bradshaw and Gee (1960) over moderate Reynolds numbers, but gives larger skin frictions at higher Reynolds numbers.

Error propagation and conditioning analysis of DNS data of turbulent viscoelastic channel flows

In the context of drag reduction phenomenon, direct numerical simulation (DNS) of turbulent viscoelastic flows is a very useful tool able to provide information that is not possible with experiments and that can be used for modeling purposes. A rare and challenging task is to provide measures for statistical errors that are present in DNS data. In the present communication, we employ a method to quantify the statistical convergence of the DNS turbulent viscoelastic channel flows from the unbalance on the mean momentum equation. In addition, we conduct a conditioning analysis of the resultant numerical linear system of this problem. We compare results from different research groups. We found that DNS databases of viscoelastic flows can induce high error propagation in the mean velocity. As a consequence, larger sampling times are needed to achieve a statistically converged Reynolds stress tensor. The conditioning analysis has shown that elasticity enhances conditioning. • We discuss the DNS importance for turbulence modeling. • Uncertainty and statistical error are analyzed in the DNS context. • Equations for mean viscoelastic turbulence (VT) are ill-conditioned. • Error propagation in the mean velocity field related to VT is high. • Statistical quantities, like Reynolds stress, in VT need to be more converged.

Layered structure of turbulent natural convection over a vertical flat plate

The layered structure of turbulent natural convection over a vertical flat plate (TNCVP) is first investigated qualitatively by dimensional analysis. The scaling patch approach is then applied to quantify the layered structure of TNCVP based on the characteristics of the mean momentum and energy equations. Fluid flow in TNCVP is divided into two layers: an inner layer and an outer layer, but heat transport is divided into four layers: a molecular diffusion sub-layer, a gradient balance layer, a meso layer and an outer layer. Scaling patch analysis identifies proper scales for the length, velocity, temperature, Reynolds shear stress, and turbulent heat flux in different layers, and develops a multi-scaling of the mean momentum and energy equations. Part of the novelty of the approach is that different regions incorporate unique scaling lengths, temperatures, turbulence temperature flux, and Reynolds stresses. The inner and outer scaled mean velocity, temperature, Reynolds shear stress and turbulent temperature flux profiles at four streamwise locations in the experiments of Tsuji and Nagano [1,2] merge onto a single curve in the inner and outer layers, respectively. This behavior supports the validity of the analysis. The layered structure of TNCVP is compared with three canonical turbulent flows: differentially heated turbulent flow in a vertical channel (DHVC), forced convection in turbulent boundary layer flows (TBL), and turbulent plumes. Striking similarities and distinctive differences among these turbulent flow and heat transport situations are elucidated. The inner layer of the mean momentum equation of TNCVP is similar to the inner layer of DHVC, but the outer layer of TNCVP is similar to that of a turbulent planar plume. The four layer structure for the mean energy equation of TNCVP is similar to that of pressure- or shear-driven turbulent boundary layers.

Large-scale magnetic field saturation and the Elsasser number in rotating spherical dynamo models

ABSTRACT Numerical simulations are used to investigate large-scale (mean) magnetic field generation in rotating spherical dynamos. Beyond a certain threshold, we find that the magnitude of the mean magnetic field becomes nearly independent of the system rotation rate and buoyancy forcing. The analysis suggests that this saturation arises from the Malkus-Proctor mechanism in which a Coriolis-Lorentz force balance is achieved in the zonal component of the mean momentum equation. When based on the large-scale magnetic field, the Elsasser number is near unity in the saturated regime. The results show that the large and small magnetic field saturate via distinct mechanisms in rapidly rotating dynamos, and that only the axisymmetric component of the magnetic field appears to follow an Elsasser number scaling.

Mean thermal energy balance analysis in differentially heated vertical channel flows

Direct numerical simulations of differentially heated vertical channel (DHVC) flows were performed for Ra=105–109 to investigate the characteristics of the streamwise mean momentum and mean thermal energy equations. The log law for mean temperature was observed for Ra≥108 at y+>50, where y+ is the wall-normal distance normalized by the viscous wall unit. From the mean momentum equation, negligible viscous force and logarithmically increasing Reynolds shear stress were observed in the region where the log law for mean temperature occurred. The streamwise mean velocity did not exhibit a linear relationship with y+ close to the wall and did not show logarithmic development far from the wall due to the buoyancy force. In the mean thermal energy equation, a constant heat flux layer was observed, and the turbulent heat flux contribution was scaled by the inverse of wall-normal distance to satisfy the log law of mean temperature. For a high Rayleigh number (Ra=109), the turbulent heat flux spectra contained scale-separated inner and outer sites with linearly growing energetic structures along the wall-normal distance, which was not observed for a low Rayleigh number (Ra=106). The flow structures of turbulent heat flux originated from the upward wall-normal velocity fluctuations that triggered the non-directional structures of the temperature. These results suggest that the scale separation between the viscous and outer length scales with the wall-attached energetic structures resulted in the log law for mean temperature. These findings could serve as the basis of scaling formulations for the mean velocity and temperature in DHVC flows.

A pressure-driven atmospheric boundary layer model satisfying Rossby and Reynolds number similarity

Abstract. Idealized models of the atmospheric boundary layer (ABL) can be used to leverage understanding of the interaction between the ABL and wind farms towards the improvement of wind farm flow modeling. We propose a pressure-driven one-dimensional ABL model without wind veer, which can be used as an inflow model for three-dimensional wind farm simulations to separately demonstrate the impact of wind veer and ABL depth. The model is derived from the horizontal momentum equations and follows both Rossby and Reynolds number similarity; use of such similarity reduces computation time and allows rational comparison between different conditions. The proposed ABL model compares well with solutions of the mean momentum equations that include wind veer if the forcing variable is employed as a free parameter.

Integral properties of turbulent natural convection over a vertical flat plate

This paper investigates global integrals of the volumetric, momentum and temperature fluxes for turbulent natural convection over a vertical plate (TNCVP). The global integral of the mean continuity equation gives a relation for the mean wall-normal velocity outside the boundary layer as W∞ = − d(∫0∞Udz)/dx where U is the mean velocity in the streamwise x direction and z is the wall-normal direction. The global integral of the mean momentum equation gives a relation for the specific wall momentum flux as uτ2 = gβ∫0∞(Θ∞ − Θ)dz − d(∫0∞U2dz)/dx where uτ is the friction velocity, g is the gravitational acceleration and β is the thermal expansion coefficient. Θ = Tw − T is the difference between the wall temperature Tw and the fluid temperature, and Θ∞ = Tw − T∞ is the difference between the wall temperature and the ambient fluid temperature. Finally, the global integral of the mean heat equation gives a relation for the wall temperature flux as uτθτ = − W∞Θ∞ − d(∫0∞UΘdz)/dx where θτ is the frictional temperature. Based on the global integrals of the momentum and heat fluxes, we propose an indirect determination of uτ and θτ. Given the extreme challenge in direct measurement of uτ and θτ in wall-bounded turbulent flows, the indirect determination proposed here offers a powerful tool to subsequent experimental studies of TNCVP.

Scaling patch analysis of turbulent planar plume

Proper scaling in turbulent planar plumes is investigated here using a scaling patch approach. Based on the scaled boundary conditions, a proper velocity scale for the mean axial flow is the plume centerline velocity Uref=Uctr, and a proper temperature scale for the temperature excess is Θref=Tctr−T∞, where Tctr is the plume centerline temperature and T∞ is the ambient fluid temperature. By seeking an admissible scaling, a key concept in the scaling patch approach, for the mean continuity, mean momentum, and mean energy equations, respectively, the following is found: (1) a proper scale for the mean transverse flow is Vref=(dδ/dx)Uctr, where dδ/dx is the growth rate of the plume width. (2) A proper scale for the Reynolds shear stress is Rvu,ref=UctrVref=(dδ/dx)Uctr2, a mix of the scales for the mean axial and transverse flows. (3) A proper scale for the turbulent heat flux is Rvθ,ref=VrefΘctr, a mix of the scales for the mean transverse flow and mean temperature excess. The mean transverse flow thus plays a critical role in the scaling of turbulent planar plumes. Approximate functions are developed for the scaled mean transverse flow, Reynolds shear stress, and turbulent temperature flux, and are found to agree favorably with experimental and numerical simulation data. The integral analysis of the mean momentum equation yields a Richardson number Ri, which remains invariant in the axial direction. The Richardson number is defined as Ri=defgβΘctrδt/(UctrVref)≈1/2, where g is the gravitational acceleration, β is the thermal expansion coefficient, and δt is the plume half-width based on the mean temperature profile. This Richardson number arises directly from the scaling patch analysis of the mean momentum equation, including both the streamwise and transverse velocity scales.

Effects of Wave–Current Interactions on Bay–Shelf Exchange

AbstractBay–shelf exchange is critical to coastal systems because it promotes self-purification or pollution dilution of the systems. In this study, the effects of wave–current interactions on bay–shelf exchange are explored in a micromesotidal system—Daya Bay in southern China. Waves can enlarge the shear-induced seaward transport and reduce the residual-current-induced landward transport, which benefits the bay–shelf exchange; however, tides work oppositely and slow the wave-induced bay–shelf exchange through vertical mixing and reduced shear-induced exchange. Five wave–current interactions are compared, and it is found that the depth-dependent wave radiation stress (WRS) contributes most to the bay–shelf exchange, followed by the wave dissipation as a source term in the turbulence kinetic energy equation, and the mean current advection and refraction of wave energy (CARWE). The vertical transfer of wave-generated pressure to the mean momentum equation (also known as the form drag) and the combined wave–current bottom stress (CWCBS) play minor roles in the bay–shelf exchange. The bay–shelf exchange is faster under southerly wind than under northerly wind because the bay is facing southeast; synoptic events such as storms enhance the bay–shelf exchange. The CARWE terms are dominant in both seasonal and synoptic variations of the bay–shelf exchange because they can considerably change the distribution of significant wave height. The WRS changes the bay–shelf exchange mainly through altering the flow velocity, whereas the wave dissipation on turbulence alters the vertical mixing. The form drag and the CWCBS have little impact on the bay–shelf exchange or its seasonal and synoptic variations.

Swirling turbulent pipe flows: Inertial region and velocity–vorticity correlations

Swirling pipe flows are studied here with an aim towards understanding the onset of the inertial region — where the turbulent-inertia term in the mean momentum equation is balanced by pressure gradient and viscous term is sub-dominant — as well as the clarifying the velocity–vorticity correlations that make up the turbulent inertia. To this end, we first manipulate the mean momentum equation in both axial and azimuthal directions and find some exact results in the inertial region, and carry out direct numerical simulations of swirling pipe flows at axial friction Reynolds numbers of 170 and 500. The swirl number considered in our simulations is S≈0.3, and we compare our results to non-swirling pipe flows at similar Reynolds numbers. We find that swirling produces a drag increase and an influence on the turbulence statistics similar to increasing the Reynolds number except for the streamwise turbulence intensity. An analysis on the axial and azimuthal mean momentum equations shows that swirling shifts the beginning of the inertial region wall-normal location closer to the wall. The turbulent inertia decomposition reveals that the near-wall region the velocity–vorticity correlations of the axial direction are similar to a 2D channel flow and interpreted as vorticity stretching/reorientation and dispersion, whereas in the new correlations in the azimuthal direction can also be given a similar physical meaning in the near-wall region. In the outer-region, however, the pipe axial correlations are different to the 2D-channel, and so are the azimuthal correlations. We find that the pipe has new a ‘geometric’ contribution in both axial and azimuthal directions that play an important role in contributing towards vorticity dispersion in the outer core region of a swirling pipe flow.