Triple Decomposition Research Articles

Experiments on two-phase flow in hydraulic jump on pebbled rough bed: Part 1–Turbulence properties and particle chord time and length

This study reported and discussed turbulence characteristics, such as turbulence intensity, correlation time scales, and advective length scales. The characteristic air–water time scale, including the particle chord time and length and their probability density functions (PDFs), was investigated. The results demonstrated that turbulence intensity was relatively greater on a rough bed in the roller length, whereas further downstream, the decay rate was higher. In addition, the relationship between turbulence intensity and dimensionless bubble count rate reflected an increase in turbulence intensity associated with the number of entrained particles. Triple decomposition analysis (TDA) was performed to determine the contributions of slow and fast turbulent components. The TDA results indicated that, regardless of bed type and inflow conditions, the sum of the band-pass (Tu′) and high-pass (Tu″) filtered turbulence intensities was equal to the turbulence intensity of the raw signal data (Tu). Tu″ highlighted a higher turbulence intensity and larger vorticities on the rough bed for an identical inflow Froude number. Additional TDA results were presented in terms of the interfacial velocity, auto- and cross-correlation time scales, and longitudinal advection length scale, with the effects of low- and high-frequency signal components on each highlighted parameter. The analysis of the air chord time indicated an increase in the proportion of small bubbles moving downstream. The second part of this research focused on the basic properties of particle grouping and clustering.

Mixing and Overturning Across the Brazil‐Malvinas Confluence

AbstractThe rates of isopycnal stirring and water mass transport by mesoscale eddies, and of diapycnal mixing by small‐scale turbulence, across the Brazil‐Malvinas Confluence (BMC) are assessed from a set of microstructure and hydrographic measurements in the Argentine Basin. This assessment is founded on a theoretical framework that applies a triple decomposition to the temperature variance equation and assumes eddies to transfer potential vorticity downgradient. The BMC is found to host widespread intense isopycnal stirring at rates of O(103–104 m2 s−1), and generally weak diapycnal mixing at rates of O(10−6–10−5 m2 s−1). Despite such disparity, both diapycnal mixing and isopycnal stirring play roles of comparable importance in determining regional water mass properties within surface and mode waters. In deeper layers, isopycnal stirring prevails. Eddies are further diagnosed to effect an important cross‐BMC transport, at rates of O(1 m2 s−1). When scaled by the along‐stream extent of the BMC, these rates integrate to volume transports that may be as large as O(10 Sverdrups). This suggests that cross‐BMC transfers of waters are substantially effected by eddy‐induced flows.

Selection of vortex ripple dimensions in sinusoidal oscillatory flows. Part 1. Ripple dimensions and fluid kinematics

Subaqueous vortex ripples in equilibrium are characterized by their unique geometry and dimensions. Motivated by the recent direct numerical simulation study of oscillatory turbulent flow over a wavy bottom by Önder & Yuan (J. Fluid Mech., vol. 858, 2019, pp. 264–314), the objective of this study is to further investigate the fluid dynamical controls that determine the distinctive equilibrium dimensions of vortex ripples. We use direct numerical simulations to investigate the differences in flow kinetics between sinusoidal oscillatory flow over equilibrium and out-of-equilibrium vortex ripples. In comparison with the equilibrium case, the spanwise coherent vortices, the averaged bottom shear stress on overlying flow and the shear stress distribution on the ripple surface are identified as the key fluid dynamical controls on equilibrium dimensions. Based on these controls, we propose mechanisms in the selection of vortex ripple dimensions. We observe that the flow adjusts in such a way that the interaction between overlying flow and vortex ripples tends to generate the strongest coherent vortices while the ripple surface (or overlying flow) experiences the smallest shear stress averaged over ripple wavelength during the selection process. Through a triple decomposition of the flow, the component of the ripple-induced fluctuation is found to dictate these fluid dynamical controls, which implies that this component plays an important role in the evolution of vortex ripples.

Structure function tensor equations with triple decomposition

Exact budget equations are derived for the coherent and stochastic contributions to the second-order structure function tensor. They extend the anisotropic generalised Kolmogorov equations (AGKE) by considering the coherent and stochastic parts of the Reynolds stress tensor, and are useful for the statistical description of turbulent flows with periodic or quasi-periodic features, like, for example, the alternate shedding after a bluff body. While the original AGKE describe production, transport, inter-component redistribution and dissipation of the Reynolds stresses in the combined space of scales and positions, the new equations, called $\varphi$ AGKE, contain the phase $\varphi$ as an additional independent variable, and describe the interplay among the mean, coherent and stochastic fields at the various phases. The newly derived $\varphi$ AGKE are then applied to a case where an exactly periodic external forcing drives the flow: a turbulent plane channel flow modified by harmonic spanwise oscillations of the wall to reduce drag. The phase-by-phase action of the oscillating transversal Stokes layer generated by the forcing on the near-wall turbulent structures is observed, and a detailed description of the scale-space interaction among mean, coherent and stochastic fields is provided thanks to the $\varphi$ AGKE.

The triple decomposition of the velocity gradient tensor as a standardized real Schur form

The triple decomposition of a velocity gradient tensor provides an analysis tool in fluid mechanics by which the flow can be split into a sum of irrotational straining flow, shear flow, and rigid body rotational flow. In 2007, Kolář formulated an optimization problem to compute the triple decomposition [V. Kolář, “Vortex identification: New requirements and limitations,” Int. J. Heat Fluid Flow 28, 638–652 (2007)], and more recently, the triple decomposition has been connected to the Schur form of the associated matrix. We show that the standardized real Schur form, which can be computed by state of the art linear algebra routines, is a solution to the optimization problem posed by Kolář. We also demonstrate why using the standardized variant of the real Schur form makes computation of the triple decomposition more efficient. Furthermore, we illustrate why different structures of the real Schur form correspond to different alignments of the coordinate system with the fluid flow and may, therefore, lead to differences in the resulting triple decomposition. Based on these results, we propose a new, simplified algorithm for computing the triple decomposition, which guarantees consistent results.

Compressed Channel Estimation for Near-Field XL-MIMO Using Triple Parametric Decomposition

With the presence of the conventional angular-domain representation, the energy leakage effect is evident for the near-field extremely large-scale multiple-input-multiple-output (XL-MIMO) channel, which further leads to ambiguous channel power detection, significantly eroding channel estimation performance. To tackle this challenge, this paper proposes a novel Triple Parametric Decomposition (TPD) framework to facilitate compressed channel estimation in the XL-MIMO system. Specifically, we examine the case of a uniform planar array (UPA) with the geometry parameters involving a pair of azimuth-elevation angles and distances in an attempt to achieve independent decomposition by constructing their own covariance matrices. Then, by utilizing the on- and off-grid techniques, we obtain the sparse recovery of angles and distances information in the absence of energy leakage effect at reduced gridding complexity. Simulation results demonstrate that our proposed TPD achieves significant performance improvement over the state-of-the-art methods.

Turbulent secondary flows in low aspect ratio open-channels over heterogeneous surfaces: Anisotropy in forces

Studies of turbulent flows over heterogeneous surfaces revealed elevated turbulent kinetic energy and Reynolds shear stress in low-momentum-path regions. These regions induce large-scale multi-cellular secondary flows. The aim of the current study is to analyze the influence of these regions on drag, lift, and lateral forces acting on spherical particles at different exposure levels, thereby addressing the hitherto unknown contribution of the spanwise inhomogeneities. For this reason, numerical simulations of turbulent open-channel flow with varying aspect ratio (AR=1,3,5) over single-sized spherical particles with diameter D were studied. Ensemble-averaged cross-flow velocity vectors showed large-scale secondary flows to penetrate in-between the spherical particles, therefore stretching over the entire flow depth. Their magnitude above 0.8D was observed to range between 12.9%and14.9% of U. Strong tertiary vortices in the vicinity of the lateral walls were identified by analysis of swirl strength. Triple decomposition of streamwise velocity fluctuations showed strong backflow at the trailing edge of the spherical particles in high-momentum-path (HMP) regions. Furthermore, it was found that drag forces are higher in HMPs, which is attributed to the larger streamwise pressure gradient.

Near wake flow and forced convection heat transfer of sinusoidal wavy cylinder based on flow decomposition

Based on flow decomposition we investigate the near wake flow and forced convection heat transfer around sinusoidal wavy cylinders, with an attempt to deepen our understanding of inherent correlations between near wake flow and heat transfer/transport. Large eddy simulations (LES) are conducted at a Prandtl number Pr = 0.7 and a subcritical Reynolds number Re = 3.0 × 103 for the unsteady flow of the sinusoidal wavy cylinder of different wavelengths λ (= 1.89, 3.79 and 6.06Dm) and a fixed wave amplitude a (= 0.152Dm), where Dm is the mean diameter of the cylinder. The near wake flow of the wavy cylinder, together with heat transport, is decomposed via the traditional triple decomposition and a newly-proposed quadruple decomposition, taking into account dominant periodic vortex shedding and/or spanwise heterogeneity of the flow. While heat transfer from the surface of the wavy cylinder to the surrounding flow is distinctly different from that of the smooth cylinder, results from the flow decomposition reveal intrinsic characteristics of the near wake flow and heat transport for the wavy cylinder with different wavelengths. It is observed that heat transport in the near wake of the wavy cylinder is predominantly associated with the coherent spanwise vortices of dominant shedding frequency as well as the spanwise time-invariant components of the mean flow. It is also found that the major contributor to Reynolds stresses (heat flux) is the temporal random stresses (temporal random heat flux) in the near wake of the wavy cylinder. Furthermore, results from the quadruple decomposition show that the spanwise-averaged heat flux in the near wake of the smooth cylinder and the wavy cylinder with λ = 1.89Dm encompasses both spatiotemporal random and temporal dispersive heat fluxes outside the recirculation bubble and only the former heat flux inside the recirculation bubble. However, the spanwise-averaged heat flux in the near wake of the wavy cylinder with λ = 3.79 or 6.06Dm is profoundly and exclusively contributed by the spatial dispersive heat flux outside the recirculation bubble.

Large- N expansion and string theory out of equilibrium

We analyze the large-$N$ expansion of general nonequilibrium systems with fluctuating matrix degrees of freedom and $SU(N)$ symmetry, using the Schwinger-Keldysh formalism and its closed real-time contour with a forward and backward component. In equilibrium, the large-$N$ expansion of such systems leads to a sum over topologies of two-dimensional surfaces of increasing topological complexity, predicting the possibility of a dual description in terms of string theory. We extend this argument away from equilibrium and study the universal features of the topological expansion in the dual string theory. We conclude that in nonequilibrium string perturbation theory, the sum over world sheet topologies is further refined: Each world sheet surface $\mathrm{\ensuremath{\Sigma}}$ undergoes a triple decomposition into the part ${\mathrm{\ensuremath{\Sigma}}}^{+}$ corresponding to the forward branch of the time contour, the part ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{-}}$ on the backward branch, and the part ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ that corresponds to the instant in the far future where the two branches of the time contour meet. The sum over topologies becomes a sum over the triple decompositions. We generalize our findings to the Kadanoff-Baym time contour relevant for systems at finite temperature and to the case of closed and open, oriented or unoriented strings. Our results are universal and follow solely from the features of the large-$N$ expansion without any assumptions about the world sheet dynamics.

Keldysh rotation in the large- N expansion and string theory out of equilibrium

We extend our study of the large-$N$ expansion of general nonequilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields ${M}_{\ifmmode\pm\else\textpm\fi{}}$---defined as the values of $M$ on the forward and backward segments of the closed time contour---for their linear combinations ${M}_{\mathrm{cl}}$ and ${M}_{\mathrm{qu}}$, known as the ``classical'' and ``quantum'' fields. First we develop a novel ``signpost'' notation for nonequilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ corresponding to the forward and backward segments of the closed time contour, and ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ of the dual string theory undergoes a very different natural decomposition: $\mathrm{\ensuremath{\Sigma}}$ consists of a ``classical'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{cl}}$ and a ``quantum embellishment'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{qu}}$. We show that both parts of $\mathrm{\ensuremath{\Sigma}}$ carry their own independent genus expansion. The nonequilibrium sum over world-sheet topologies is naturally refined into a sum over the double decomposition of each $\mathrm{\ensuremath{\Sigma}}$ into its classical and quantum part. We apply this picture to the classical limits of the quantum nonequilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.

On the scattering of entropy waves at sudden area expansions

In this work, we investigate both numerically and theoretically the sound generated by entropy waves passing through sudden area expansions. This is a canonical configuration representing internal flows with flow separation and stagnation pressure losses. The numerical approach is based on a triple decomposition of the flow variables into a steady mean, a small-amplitude coherent part, and a stochastic turbulent part. The coherent part contains acoustic, vortical, and entropy waves. The mean flow is obtained as the solution of the Reynolds-Averaged Navier–Stokes (RANS) equations. The equations governing the coherent perturbations are linearised and solved in the frequency domain. To account for the effect of turbulence on the coherent perturbations, a frozen eddy viscosity model is employed. When entropy fluctuations pass through the area expansion, the generated entropy noise behaves as a low-pass filter. The numerical predictions of the noise at low frequencies are compared to the predictions of compact, quasi-one-dimensional, and isentropic theory and large discrepancies are observed. An alternative model for the generated entropy noise tailored for area expansions is then proposed. Such model is based on the conservation of mass, momentum, and energy written in integral form. The model assumes zero frequency and the one-dimensionality of the flow variables far upstream and downstream of the expansion. The predictions of this model agree well with the numerical simulations across a range of finite subsonic Mach numbers including low, intermediate, and high Mach numbers. The contributions of this work are both numerical and theoretical. Numerically, a triple decomposition adapted to high-Mach-number, compressible flows is introduced for the first time in the context of acoustic simulations. From a theoretical point of view, the quasi-steady model proposed here correctly captures the low-frequency entropy noise generated at sudden area expansions, including at high subsonic Mach numbers.

Energetics and vortex structures near small-scale shear layers in turbulence

Vortices and kinetic energy distributions around small-scale shear layers are investigated with direct numerical simulations of isotropic turbulence. The shear layers are examined with the triple decomposition of a velocity gradient tensor. The shear layers subject to a biaxial strain appear near vortices with rotation, which induce energetic flow that contributes to the shear. A similar configuration of rotating motions near the shear layers is observed in a multi-scale random velocity field, which is free from the dynamics of turbulence. Therefore, the mechanism that sustains shearing motion is embedded as a kinematic nature in random velocity fields. However, the biaxial strain is absent near the shear layers in random velocity because rotating motions appear right next to the shear layers. When a random velocity field begins to evolve following the Navier–Stokes equations, the shear layers are immediately tilted to the nearby rotating motions. This misalignment is a key for the vortex to generate the compressive strain of the biaxial strain around the shear layer. As the configuration of shearing and rotating motions arises from the kinematic nature, the shear layers with the biaxial strain are formed within a few times the Kolmogorov timescale once the random velocity field begins to evolve. The analysis with high-pass filtered random velocity suggests that this shear layer evolution is caused by small-scale turbulent motions. These results indicate that the kinematic nature of shear and rotation in velocity fluctuations has a significant role in the formation of shear layers in turbulence.

Outer-layer coherent structures from the turbulent/non-turbulent interface perspective at moderate Reynolds number

This study reports the observation of outer-layer coherent structures in turbulent boundary layer from the turbulent/non-turbulent interface perspective at moderate Reynolds number. The observation relies on time-resolved particle image velocimetry measurements with a large field-of-view. The local turbulent kinetic energy is used to detect the turbulent/non-turbulent interface. In the reference frame of turbulent/non-turbulent interface, raw velocity is decomposed into three components: time average of raw velocity, conditionally averaging components and turbulent fluctuations. The triple velocity decomposition eliminates the bias of Reynolds decomposition fluctuations in the intermittent region. The conditional statistics based on the turbulent/non-turbulent interface is considered to discuss the turbulence characteristics. The basic statistics of mean raw velocity, Reynolds stresses, and swirling strength are observed in conditional statistics. And the results are compared with the results based on the traditional wall surface reference frame. Quadrant analysis is introduced to extract ejections & sweeps and reflect the statistical characteristics of the outer-layer structure. The contribution of ejections and sweeps to the Reynolds shear stresses exhibits significantly different characteristics. Then, the configuration of the turbulent fluctuations and instantaneous structures is presented relied on the uniform momentum zones in the instantaneous frames. The spatial characteristics of coherent structures are reflected by two-point correlation of the streamwise turbulent fluctuations. In addition, the spatial scale of the outer-layer structures is revealed from the turbulent/non-turbulent interface perspective.

Direct numerical simulations of turbulent flow over misaligned traveling waves

Direct numerical simulations of turbulent flow over prescribed traveling waves in a half-channel flow setup subject to a combined streamwise and spanwise pressure gradient are performed at a friction Reynolds number of 180. The simulations undertaken in this study consider both flow-aligned (including opposing and following) and misaligned waves as we attempt to quantify the effect of wave misalignment on turbulence statistics, the pressure drag force, and the wave attenuation rate. For the simulations, we consider three characteristic wave age values corresponding to slow-, intermediate-, and fast-moving waves. Wave misalignment is taken into account by applying a spanwise pressure gradient vertical to the traveling waves, which results in a three-dimensional turbulent flow field above the moving waves. Key flow quantities such as the mean velocity, velocity variances, and momentum fluxes are found to vary with the wave parameters, confirming the findings of previous studies. In addition, our analysis shows that the mean velocity vector deviates from the applied pressure gradient with larger discrepancies being found when fast-moving waves run in an obtuse angle of 135° relative to the applied pressure gradient vector. Additionally, large deviation angles (up to 50°) between the applied pressure gradient and the shear-stress vector were found to exhibit their peak within the buffer layer in ζ+<10, whereas they rapidly reduce for ζ+>10, where ζ+ is the vertical coordinate in wall units. Finally, with the help of triple decomposition we show the dependence of the spanwise wave-coherent velocity on the dynamics of the vertical component and highlight important differences between flow-following and flow-opposing waves.

Hybrid RANS/LES study of tip leakage vortex instability and turbulence characteristics of a transonic turbine cascade

To conduct detailed investigations about vortex instability and turbulence characteristics near a transonic turbine tip region, a highly accuracy hybrid Reynolds averaged Navier Stokes (RANS) / large eddy simulation (LES) method, namely detached eddy simulation (DES)-type method, is adopted at the exit isentropic Mach number of 0.95. Validation results show that this hybrid method can provide an accurate prediction against experiments. In contrast with the RANS method, this method can exhibit superiority in capturing detailed small-scale flow structures and more realistic turbulence distribution. Based on various identification methods, two predominant instability patterns of tip leakage vortex (TLV), namely, vortex wandering and breakdown have been recognized successfully in its downstream progression. Due to the supplement of tip leakage flow (TLF), the TLV maintains a concentrated vortex morphology with a monotonically increasing transverse wandering magnitude inside the blade passage. Subsequently, the TLV breakdown phenomenon, which is mainly due to the high adverse pressure gradient aggravated by shock waves, can be observed downstream the trailing edge. Meanwhile, a “spiral-type” vortex breakdown is distinguished firstly in a transonic turbine environment, which cannot be found in the previous RANS literature. To investigate the nature of the vortex instability, we perform an analysis based on the vorticity transport equation, it's found that the convection term due to the velocity gradient dominated the leakage vorticity damping. Combined the triple decomposition with the proper orthogonal decomposition (POD) method, the mechanisms of vortex wandering have been further revealed, which exhibits non-zero perturbations in transverse modes and a helical mode in the streamwise direction. Furthermore, detailed analysis of the turbulence characteristics has been also conducted, which contains distributions of Reynolds stress terms and the turbulent kinetic energy (k) along the TLV core trajectory. It can be recognized that the high adverse pressure gradient by shock waves can result in a sharp increase of Reynolds normal stresses (τii), which can stretch the TLV in different directions and exerts detrimental effects on the vortex breakdown. As a complementary, the turbulent anisotropy is also evaluated by the Lumley triangle method, which exhibits a variation from the one-component anisotropy to a two-component morphology as the TLV migrating downstream, while a isotropic turbulence field may be formed downstream the trailing edge due to the intense mixing. Predicted turbulence characteristics, vortical instabilities, and their flow mechanism can provide guide towards lower loss design and flow control of the turbine blade.

Analysis of Large Eddy Simulations and 1D Hot-Wire Data to Determine Actively Generated Main Flow Turbulence in a Film Cooling Test Rig

Abstract Active turbulence generators were incorporated into a wind tunnel to investigate more realistic inflow conditions for a film cooling test rig. The flow field signals are sampled numerically by probes in large eddy simulations (LES) and experimentally using 1D hot-wire measurements to determine turbulence quantities. The LES shows that the turbulence is anisotropic which cannot be detected by the 1D hot-wire. Furthermore, the integral length scale which shall provide insight into the sizes of the turbulent eddies is determined using two approaches. The first uses the one probe at two times correlation method and could be evaluated from the numerical and experimental probes. The second correlation method exploits the spatial resolution in the LES domain using the two probes at one time approach. Both methods show combustor-like turbulence length scales downstream of the active turbulence generators if the triple decomposition is applied onto the velocity signal.

Disappearing vortex problem in vortex identification: Non-existence for selected criteria

A discontinuous outcome of vortex-identification methods called the disappearing vortex problem (DVP) has been already found for the swirling strength criterion and the Rortex (later renamed as Liutex) method. Here, the opposite property reflecting the situation that the DVP cannot be found for any input data, that is, the non-existence of the DVP, is examined and proved valid for selected criteria based on the velocity-gradient tensor, including Q, lambda-2, and the triple decomposition method. For the Q-criterion and the triple decomposition method, it is done directly, whereas for lambda-2, it is shown using a proof by contradiction.

Computational Analysis of Flow Structures in Turbulent Ventricular Blood Flow Associated With Mitral Valve Intervention.

Cardiac disease and clinical intervention may both lead to an increased risk for thrombosis events due to a modified blood flow in the heart, and thereby a change in the mechanical stimuli of blood cells passing through the chambers of the heart. Specifically, the degree of platelet activation is influenced by the level and type of mechanical stresses in the blood flow. In this article we analyze the blood flow in the left ventricle of the heart through a computational model constructed from patient-specific data. The blood flow in the ventricle is modelled by the Navier-Stokes equations, and the flow through the mitral valve by a parameterized model which represents the projected opening of the valve. A finite element method is used to solve the equations, from which a simulation of the velocity and pressure of the blood flow is constructed. The intraventricular blood flow is complex, in particular in diastole when the inflow jet from the atrium breaks down into turbulent flow on a range of scales. A triple decomposition of the velocity gradient tensor is then used to distinguish between rigid body rotational flow, irrotational straining flow, and shear flow. The triple decomposition enables the separation of three fundamentally different flow structures, that each generates a distinct type of mechanical stimulus on the blood cells in the flow. We compare the results in a simulation where a mitral valve clip intervention is modelled, which leads to a significant modification of the intraventricular flow. Further, we perform a sensitivity study of the results with respect to the positioning of the clip. It was found that the shear in the simulation cases treated with clips increased more compared to the untreated case than the rotation and strain did. A decrease in valve opening area of 64% in one of the cases led to a 90% increase in rotation and strain, but a 150% increase in shear. The computational analysis opens up for improvements in models of shear-induced platelet activation, by offering an algorithm to distinguish shear from other modalities in intraventricular blood flow.

Development of turbulent heat flux model for unsteady forced convective heat transfer of small-to-medium Prandtl-number fluids based on deep learning

Turbulent heat flux (THF) models are used for the closure of the THF term when solving the steady/unsteady Reynolds-averaged scalar transport equation to simulate the turbulent heat transfer in industry. It is known that the simple gradient diffusion hypothesis (SGDH) has deficiencies under complex conditions with flow separations. To develop a more general THF model, this paper firstly establishes a high-fidelity database of forced convective heat transfer passing a circular cylinder under different Prandtl number (Pr) conditions at the Reynolds number (Re) of 500 via the direct numerical simulations. Proper orthogonal decomposition method is then employed for the triple decomposition on unsteady turbulent flow with the first two orders of eigenmodes reconstructing the large-scale field and the remaining reconstructing the turbulent field. Then, architectures using different neural network structures based on tensor basis neural network (TBNN), including MLP-TBNN-THF which adopts the multilayer perceptron (MLP) and ResNet-TBNN-THF that uses the residual network (ResNet), are constructed to predict normalized THF from large-scale flow features and Pr. Posterior tests are carried out on different Prs and three different Res: 500, 5000 and 16900 using the well-trained TBNN-THF models to evaluate their performance and generalization capability. Most models proposed in this paper predict the heat transfer more accurately than the SGDH model, even when extended to conditions out of the range trained. The failure of the isotropic assumption of SGDH model is observed in most regions. In that case, it is vitally necessary to use a model like proposed in this paper to simulate the THF term more accurately. In conclusion, the THF model, which is appropriate for complex working conditions and fluids with small-to-medium range of Pr, obtained by deep learning method in this paper, is helpful to improve the prediction accuracy of temperature field or concentration field by steady or unsteady Reynolds-averaged Naiver-Stokes approach in engineering. Besides, the current framework can be generalized to scalar-flux modeling under other conditions with the supplement of more databases in the future.

Decomposition of the Reynolds shear stress in a turbulent boundary layer modified by miniature vortex generators

Spanwise modification of the Reynolds shear stress (RSS) of a turbulent boundary layer modified by miniature vortex generators (MVGs) is investigated. The methodology is based on quadrant analysis of RSS to first identify the spatial and temporal information of the RSS. We then apply the Fourier mode decomposition and triple velocity decomposition to obtain various statistics of the decomposed RSS. In addition, the interactions between small-scale and large-scale RSS are further investigated and we showed that the large scales tend to strengthen due to the inverse energy transfer of small scales.