Constant Eddy Viscosity Research Articles

Eddy viscosity enhanced temporal direct deconvolution model for temporal large-eddy simulation of turbulent channel flow

In this work, a novel eddy viscosity enhanced temporal direct deconvolution model (TDDM) is proposed for temporal large-eddy simulation (TLES) of turbulent channel flow at large filter widths. To improve the accuracy of the constant eddy viscosity (CEV) model, particularly in the near-wall region, a damping function is incorporated to refine its performance. Moreover, a spatial filtering strategy is introduced to reduce the aliasing errors associated with the computation of subfilter-scale (SFS) stress, thereby enhancing numerical stability. In the a posteriori study, the accuracy of the CEV model is assessed comprehensively by comparing the TLES results with corresponding temporally filtered direct numerical simulation data. The results demonstrate that the CEV-enhanced TDDM provides accurate predictions across various statistical properties of velocity, instantaneous flow structures, kinetic energy spectra, and SFS energy fluxes. The coefficient sensitivity analysis of the CEV model reveals that the model coefficient significantly influences low Reynolds number flows, while its impact on high Reynolds number flows is relatively small. TLES on coarse grids demonstrate that the CEV-enhanced TDDM exhibits strong robustness and accuracy at different grid resolutions. Additionally, the CEV-enhanced TDDM in high Reynolds number flows is stable and accurate at remarkably large filter widths.

Sensitivity of boundary layer features to depth-dependent baroclinic pressure gradient and turbulent mixing in an ocean of finite depth

The present numerical study builds on Ekman (1905)’s work in surface boundary layer and extends the boundary value problem to overcome some of its limitations. Previous studies addressed model’s limitations by assuming that deviations from observations are usually ascribed to different eddy viscosity shapes, but seldom to the presence of baroclinic pressure gradients and shallow seas, which are the mainstays of this work. Improved solutions in the ocean boundary layer are obtained considering both depth-dependent wind-induced eddy viscosity and horizontal density gradients, ranging from well-mixed to highly-stratified conditions in a finite-depth ocean. High-order numerical solutions extend those in previous analytical and numerical works in the literature and widens the parameter space analyzed. Remarkably, the current profiles are obtained without ambiguity as a truly superposition of a geostrophic and a ageostrophic terms. Results indicate that, for a vertically-uniform eddy viscosity without density gradients and in shallow waters, currents are practically aligned with wind. As depth increases, misalignment between currents and wind increases and the complexity of the vertical structure increases. At large depths, Ekman’s values are attained, i.e., deflection angles relative to wind direction, θW, are θS−θW=−45∘ at the surface, where the current is maximum, and θT−θW=−90° for the depth-integrated transport (negative for deflections to the right in the Northern Hemisphere). These features remain regardless of the magnitude of the eddy-viscosity. For non-uniform eddy viscosity, θS−θW decreases from −45∘ up to −90∘ from low to high stratification level, respectively, whereas θT−θW is rather insensitive (θT−θW≈−90∘). Contrary to wind effects, the presence of the only vertically-uniform density gradient forcing, with constant eddy viscosity, deflects the surface angle θS−θD=+45∘ relative to the density-gradient direction, θD, in deep waters. Maximum currents no longer occur at the surface in this case. For non-uniform density-gradient profiles, current magnitude decreases overall while θS−θD≈+45∘ as long as the gradient affects the entire surface boundary layer. The deflection angle θT−θD∼+95∘ remains less sensitive to changes in density-gradient profile.

Effect of constant eddy viscosity assumption on optimization using a discrete adjoint method

This paper presents a thorough study of the effect of the Constant Eddy Viscosity (CEV) assumption on the optimization of a discrete adjoint-based design optimization system. First, the algorithms of the adjoint methods with and without the CEV assumption are presented, followed by a discussion of the two methods’ solution stability. Second, the sensitivity accuracy, adjoint solution stability, and Root Mean Square (RMS) residual convergence rates at both design and off-design operating points are compared between the CEV and full viscosity adjoint methods in detail. Finally, a multi-point steady aerodynamic and a multi-objective unsteady aerodynamic and aeroelastic coupled design optimizations are performed to study the impact of the CEV assumption on optimization. Two gradient-based optimizers, the Sequential Least-Square Quadratic Programming (SLSQP) method and Steepest Descent Method (SDM) are respectively used to draw a firm conclusion. The results from the transonic NASA Rotor 67 show that the CEV assumption can deteriorate RMS residual convergence rates and even lead to solution instability, especially at a near stall point. Compared with the steady cases, the effect of the CEV assumption on unsteady sensitivity accuracy is much stronger. Nevertheless, the CEV adjoint solver is still capable of achieving optimization goals to some extent, particularly if the flow under consideration is benign.

Semi-analytical method to estimate boundary shear stress in smooth rectangular and trapezoidal open channels

The boundary shear stress is obtained from the continuity and momentum equations in smooth prismatic open channels. Shear stress is a function of gravity, secondary flow, and gradient velocity. In this research, semi-analytical equations were obtained to estimate the average boundary shear stress in smooth rectangular and trapezoidal open channels by 0.5, 1, 1.5 and 2 side slopes using conformal mapping techniques. After dividing the channel cross section into the bed and wall sub-sections, the present study used the method by Guo and Julien (2005) to estimate the average shear stress of the bed and the wall in open channels. For the studied case, by the first shear stress estimation, secondary current and constant eddy viscosity were neglected. After recommending two secondary current and eddy viscosity correction factors, a second approximation well matched the experimental measurements of the average shear stress for all aspect ratios.

Eddy viscosity enhanced temporal direct deconvolution models for temporal large-eddy simulation of turbulence

A dynamic eddy viscosity (DEV) model and a constant eddy viscosity (CEV) model are proposed for stabilizing the temporal direct deconvolution model (TDDM) in temporal large-eddy simulation of turbulence. Compared to the original unresolved subfilter-scale model used in TDDM, the new eddy viscosity models reduce the number of empirical coefficients and make TDDM more convenient to be applied in practice. The DEV model does not have any empirical coefficients, and the CEV model has only one constant model coefficient that is independent of the filter width and insensitive to the grid resolution. To solve the stability issue of TDDM, an algorithm called the variable filter-width method (VFM) is proposed. In VFM, the filter width is initialized by a small value or 0 and then grows linearly in a small number of time steps until it reaches the target filter width. The three dimensional homogeneous isotropic turbulence is applied to investigate the performance of the proposed models. In the a posteriori testing at different grid resolutions, eddy viscosity enhanced temporal direct deconvolution models show a good accuracy in predicting various statistics and instantaneous spatial structures of turbulence, and they perform better than the original model, especially in the prediction of subfilter-scale (SFS) stress and SFS energy flux. Moreover, the energy spectrum and other flow statistics predicted by the CEV model with a fixed model coefficient 0.03 are in a good agreement with the filtered DNS.

Boundedness of atmospheric Ekman flows with two-layer eddy viscosity

In this paper, we study the boundedness of atmospheric Ekman flows with classical boundary conditions. We consider the system with a two-layer eddy viscosity, consisting of a constant eddy viscosity in the upper layer and a continuous eddy viscosity in the lower layer. We analyze the boundedness of the solution by using the logarithmic matrix norm.

Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver

This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated codes are manually adjusted to reduce memory and CPU time consumption. This manual adjustment is performed by replacing the automatically generated low-efficient differentiated codes with manually developed ones. To demonstrate the effectiveness of the proposed approach, the single-stage transonic compressor–NASA Stage 35 and the 1.5-stage Aachen turbine–are used. The solution information exchange at a rotor-stator/stator-rotor interface is achieved by a conservative, non-reflective, and robust discrete adjoint mixing plane method. The results show that the discrete adjoint solver developed by hybrid automatic and manual differentiation is more economical in computational cost than that developed purely by an AD tool and has higher sensitivity accuracy than the adjoint solver with the constant eddy viscosity (CEV) assumption. Moreover, the multi-row turbomachinery design optimizations can be efficiently performed by the discrete adjoint solver developed by the hybrid automatic and manual differentiation.

Influence of Stratification and Bottom Boundary Layer on the Classical Ekman Model

A depth understanding of the different processes of water movements produced by the wind surface stress yields a better description and improvement of the marine food chain and ecosystem. The classical Ekman model proposes a hypothetical ocean, excluding the influence of continents and the Coriolis force. It also assumes infinite depth and a constant vertical eddy viscosity. The current study aims to understand how the vertical velocity profile is affected by the variation of the eddy viscosity coefficient (kz) and the consideration of a finite depth. The study uses an ideal analytical model with the Ekman classical model as a starting point. It has been demonstrated that, for a very stratified profile, when the depth is not considered infinity, the Ekman transport tends to a direction smaller than 80°. It differs from the classical Ekman model, which proposes an approximated angle equal to 90°. Considering the modified model, it was also found that the surface current deviation is smaller than 40°, which differs from the 45° proposed by the classical model. In addition, it was determined that for ocean depths smaller than 180 m, the maximum velocity does not occur at the water surface, as in the classical model, but at deeper levels.

Spatial wave solutions for generalized atmospheric Ekman equations

In this paper, we use Prandtl mixing-length theory and semiempirical theory to extend the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. New generalized atmospheric Ekman equations are established and qualitative properties of the corresponding ODEs are studied. Spatial wave solutions results for the nonlinear and implicit equations with different nonlinearities are presented.

True gravity in ocean dynamics Part 1 Ekman transport

The direction normal to the Earth spherical (or ellipsoidal) surface is not vertical (called deflected vertical) since the vertical direction is along the true gravity g (= igλ+ jgφ+ kgz). Here, (λ, φ, z) are (longitude, latitude, depth), and (i, j, k) are the corresponding unit vectors. The spherical (or ellipsoidal) surfaces are not horizontal surfaces (called deflected-horizontal surfaces). The most important body force g (true gravity) has been greatly simplified without justification in oceanography to the standard gravity (-g0k) with g0 = 9.81 m/s2. Impact of such simplification on ocean dynamics is investigated in this paper using the Ekman layer model. In the classical Ekman layer dynamic equation, the standard gravity (-g0k) is replaced by the true gravity g(λ, φ, z) with a constant eddy viscosity and a depth-dependent-only density ρ(z) represented by an e-folding near-inertial buoyancy frequency. New Ekman spiral and in turn new formulae for the Ekman transport are obtained for ocean with and without bottom. With the gravity data from the global static gravity model EIGEN-6C4 and the surface wind stress data from the Comprehensive Ocean-Atmosphere Data Set (COADS), large difference is found in the Ekman transport using the true gravity and standard gravity.

Retracted: True Gravity in Atmospheric Ekman Layer Dynamics

AbstractTrue gravity is a three‐dimensional vector field, g(λ, φ, z) = igλ + jgφ + kgz, with (λ, φ, z) the (longitude, latitude, height) and (i, j, k) the corresponding unit vectors. The longitudinal‐latitudinal component of the true gravity, gh = igλ+jgφ, is neglected completely in meteorology through using the standard gravity (−g0k, g0 = 9.81 m/s2) or the effective gravity [−g(φ)K]. Here, k (or K) is normal to the Earth spherical (or ellipsoidal) surface. Such simplification of g(λ, φ, z) has never been challenged. This study uses the classical atmospheric Ekman layer dynamics as an example to illustrate the importance of gh. The standard gravity (−g0k) is replaced by the true gravity g in the classical atmospheric Ekman layer equation with a constant eddy viscosity (K) and a height‐dependent‐only density ρ(z) represented by an e‐folding stratification. New formulas for the Ekman spiral and Ekman pumping are obtained. The second derivative of the gravity disturbance (T), , causes the Ekman pumping in addition to the geostrophic vorticity (). With from the EIGEN‐6C4 static gravity model, and calculated from July sea level pressure (p) data from the Comprehensive Ocean‐Atmosphere Data Set, the global mean strength of the Ekman pumping over the world oceans is 3.69 cm s−1 due to gh, which is much larger than 0.33 cm s−1 due to the geostrophic vorticity. It implies the urgency to use the true gravity g(λ, φ, z) into atmospheric GCM and weather forecast although the results are obtained from specific density field and gravity model.

Perturbation Analysis for the Surface Deflection Angle of Ekman-Type Flows with Variable Eddy Viscosity

In the setting of steady wind-driven oceanic currents in a non-equatorial two-layer ocean with constant eddy viscosity in the lower layer and a general continuous eddy viscosity profile in the upper layer, we derive a formula for the change of the surface deflection angle due to a perturbation (in the upper layer) of a given eddy viscosity profile for which the solution to the problem is known. This extends and generalizes previous results for perturbations of a constant eddy viscosity. The perturbation analysis is performed using a recent reformulation of the linear second-order governing equations for Ekman-type flows as a nonlinear first-order Riccati-type equation.

The Ekman spiral for piecewise-constant eddy viscosity

ABSTRACT We investigate the classical problem of wind-stress-induced non-equatorial steady ocean currents, considering a three-valued piecewise-constant eddy viscosity. This extends and generalizes recent results on piecewise-uniform eddy viscosity with only two layers. The aim is to determine how variations in the relative values of the eddy viscosity at different depths may account for the deviations from the surface current deflection of 45, predicted in the classical constant eddy viscosity case. Such variations are observed in field data, and we investigate analytically the effect the presence of a third region makes compared to the two-layer case.

Explicit solution and dynamical properties of atmospheric Ekman flows with boundary conditions

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with the constant eddy viscosity. Different from the previous work, we modify the boundary conditions and derive the explicit solution by using the notation of matrix cosine and matrix sine. For the arbitrary height-dependent eddy viscosity, we get the solution of the classical problem with zero velocity and acceleration at the bottom of the layer. In addition, uniqueness is shown and dynamical properties of solution are characterized.

On the Nature of the Turbulent Energy Dissipation Beneath Nonbreaking Waves

AbstractHere we have determined the nature of turbulent flow associated with oceanic nonbreaking waves, which are on average much more prevalent than breaking waves in most wind conditions. We found this flow to be characterized by a low turbulence microscale Reynolds number of 30 < Reλ < 100. We observed that the turbulent kinetic energy dissipation rate associated with nonbreaking waves ϵ, ranged to 3 · 10−4 W/kg for a wave amplitude 50 cm. The ϵ, under nonbreaking waves, was consistent with ; Sij is the large‐scale (energy‐containing scales) wave‐induced mean flow stress tensor. The turbulent Reynolds stress associated with nonbreaking waves was consistent with experimental data when parameterized by an amplitude independent constant turbulent eddy viscosity, 10 times larger than the molecular value. Given that nonbreaking waves typically cover a much larger fraction of the ocean surface (90–100%) than breaking waves, this result shows that their contribution to wave dissipation can be significant.

Fully-turbulent adjoint method for the unsteady shape optimization of multi-row turbomachinery

The possibility of taking into account unsteady flow effects if performing turbomachinery shape optimization is attractive to accurately address inherently time dependent design problems. The harmonic balance method is an efficient solution for computational fluid dynamics problems of turbomachinery characterized by quasi-periodic flows. If applied in combination with adjoint methods, it enables the possibility to deal with unsteady fluid-dynamic design in a cost effective manner, opening the way towards multi-disciplinary applications. This paper presents the development of a novel fully-turbulent discrete adjoint based on the time domain harmonic balance method and its application to the constrained fluid dynamic optimization of an axial turbine stage. As opposed to previous works, the proposed method does not require any assumption on the turbulent eddy viscosity and on the set of input frequencies. The results show that the method provides accurate gradients, if compared with second order finite differences, and significant deviation with respect to the sensitivity computed with the constant eddy viscosity approximation. The application of the method to the fluid-dynamic shape optimization of the exemplary stage leads to improve the total-to-static efficiency of 0.8%. The efficiency increase is found to be higher than that obtained by means of a steady state optimization method.

Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Different from the well-known homogeneous system in [ 14 , 20 ], we retain the turbulent fluxes and establish a new nonhomogeneous system of first order differential equations involving a term with the horizontal dependent. We present the existence and uniqueness of periodic solutions and show the Hyers-Ulam stability results for the nonhomogeneous systems under the mild conditions via the matrix theory. Further, we consider the nonhomogeneous systems with varying eddy viscosity coefficient and study systems with piecewise constants, systems with small oscillations, systems with rapidly varying coefficients and systems with slowly varying coefficients and give more continued results.

Eddy viscosity from bottom Ekman veering profiles

ADCP profiles near the sea bed, at a 130 m depth continental shelf, were used to estimate the mean veering of the velocity vectors using a complex correlation coefficient. An analytic expression for the veering vertical gradient (∂θ/∂z) as function of turbulent Ekman layer thickness (δE) was developed and, using constant eddy viscosity assumptions, sea bottom stress and additional turbulent boundary layer features were evaluated. Further statistics on the time series yields median values for δE of about 13 m, eddy viscosity Az = 4.2×10−3 m2s−1 and median bottom stress of 9.2×10−2 Pa, thus leading to high Reynolds numbers. Very good agreement is obtained between observations and bottom Ekman balance solutions according to determination coefficients assessed (R≥0.99).

What do we need to Probe Upper Ocean Stratification Remotely?

We consider whether it is possible in principle to retrieve the key parameters of the mixed layer in the upper ocean (its thickness, bulk eddy viscosity and the pycnocline stratification below) using a theoretical model, which assumes the surface velocity and wind stress to be known from observations. To this end we examine the dynamics of the Ekman current in the novel two-layer model of the upper ocean made of two layers with greatly differing constant eddy viscosities. The presence of stratification manifests itself through suppression of turbulence and, hence, in much smaller value of the eddy viscosity compared to the bulk eddy viscosity ve1 in the mixed layer. Within this two-layer model the general solution in terms of Green’s function has been derived and analyzed. It was found that a spectral component of frequency ω of the Ekman current on the surface “feels” the presence of the stratified layer when the mixed layer depth d is less then or comparable to the Ekman scale $$ \sqrt{2{v}_{\mathrm{e}1}/f+\omega } $$ , where f is the Coriolis parameter. Thus, under conditions of strong wind resulting in large eddy viscosity ve1, the depth of the mixed layer could be (in principle) inferred from the observations of wind and surface velocity. We conclude by stating that to retrieve the mixed layer parameters from the wind and surface velocity data, the theoretical model has to be extended by taking into account the effects of the Stokes drift due to surface waves and the possibility of intense mixing at the bottom of the mixed layer.

Turbulence models evaluation in MPS Lagrangian simulation of marine waters

ABSTRACT An advanced and well-equipped Moving Particle Semi-implicit which is a Lagrangian method is employed to simulate hydrodynamic behaviour of water wave evolution. Unlike most recently developed MPS models that consider inviscid or laminar flow, the present developed model is based on the turbulent nature of the fluid flow. In order to consider the turbulence stresses, three turbulence closures are included in the model, which are constant eddy viscosity, mixing length and k-ε model. The developed model was applied to different case studies including wave run-up on an inclined wall, wave train induced by wavemaker and landslide-generated wave. The model results were compared with analytical, empirical and experimental data cited in the literature and a good agreement was found. The results also showed that deploying a turbulence model could improve the stability of the MPS model. Results obtained through this research reveal that the application of k-ε model in combination with MPS method produces accurate and reliable results in addition to a stable solution for free surface prediction in wave mechanics.