Log-law Layer Research Articles

Effects of streamwise rotation on helicity and vortex in channel turbulence

Helicity plays a key role in the evolution of vortex structures and turbulent dynamics. The helicity dynamics and vortex structures in streamwise-rotating channel turbulence are discussed in this paper using the helicity budget equation and the differentiated second-order structure function equation of helicity. Generally, rotation and Reynolds numbers exhibit opposing effects on the interscale helicity dynamics and the vortices. Under the buffer layer, the positions of the helicity peaks are proportional to the ratio between the Reynolds and rotation numbers. The mechanism is related to the opposing effects of convection and rotation. Rotation directly affects the helicity balance through the Coriolis term and corresponding pressure term. In the buffer layer, the scale helicity is negative at small scales but positive at large scales, which is mainly induced by the spatial effects (the production and the spatial turbulent convection) but reduced by interscale cascades. Examination of structures reveals the close association between scale helicity and streaks, with streak lift angles exhibiting an increase with rotation and a decrease with Reynolds numbers. In the log-law layer, the Coriolis terms and corresponding pressure terms are proportional to the rotation numbers but remain independent of the Reynolds numbers. The negative scale helicity is forward cascaded towards small scales. Generally, spanwise vortices in the log-law layer are related to sweep events and forward cascades. Our findings indicate that these spanwise vortices are suppressed by rotation but recover with increasing Reynolds numbers, aligning with the effects observed in the scale helicity balance.

Scale effect of micro ribs on the turbulent transport in an internal cooling channel

Owing to the limited supply and pressure margin in the air system, a cooling technique providing efficient heat transfer with lower flow loss is highly desirable for gas turbine blades. Microscale ribs have promised to be a potential cooling candidate. In this work, large eddy simulations are implemented to reveal the scale effect of micro ribs on the near-wall turbulent transport in a cooling channel. Considering a mechanistic study and practical applications, both single-rib and rib-array arrangements are studied with a wide range of dimensionless viscous-scaled rib heights involving the entire boundary layer. The results indicate that the rib-induced destruction and regeneration of coherent structures are, respectively, responsible for the weakened momentum transport and enhanced heat transport in the near-wall region. Using tiny ribs, regenerated quasi-streamwise vortices are mainly located in the buffer layer. The resulting turbulence burst greatly enhances wall heat transfer while keeping a lower flow loss due to the weak form drag. Regenerated hairpin vortices using tall ribs are activated in the log-law layer and intensively interact with mainstream. Along with improved wall heat transfer, the significant form drag results in a remarkably high flow loss. Accordingly, heat transfer and flow loss show different dependencies on the rib height, which contributes to an optimum height interval of ribs (e+ = 20–40) located in the high buffer and low log-law layer for maximizing the overall performance. Furthermore, for the rib-array scheme, adequate inter-rib spacing is essential to achieve turbulence regeneration for enhancing near-wall heat transport.

Detailed turbulent structures investigation around piers group induced flow

This paper aims to investigate the mean velocities and turbulence characteristics, which include Reynolds shear stresses (RSS) and turbulent intensities, in order to study temporal and spatial flow patterns around a complex three square-piers group. Measurements have been performed by acoustic Doppler velocimeter (ADV) to identify turbulence flow structures alteration and vortices formation in longitudinal, transverse, and vertical directions. In total, 164 locations around the bridge piers have been measured, while each location consists of no lesser than 15 vertical depth ADV data point measurements. To investigate the flow turbulence, critical locations have been observed from high vortex-influenced regions. The impacted mean velocity profiles have been compared to that at the unaffected region to study the flow alteration caused by piers group. The result reveals that the log-law layer near wall has been enhanced by the installation of pier. Furthermore, as identified from the most critical locations of turbulent structures across different measurement rows, the accelerated flow will result in lower observed turbulent structures. Although flow after each pier has been influenced by strong reversal velocities, vortices have managed to form at near wake. Analysis has been divided into inner and outer flow regions (IFR and OFR) based on flow depth to enhance the understanding of how bridge piers contribute to the development of flow turbulence.

Turbulent superstructure statistics in a turbulent boundary layer with pressure gradients

In turbulent boundary layers, streamwise elongated regions of high- and low-momentum in the log-law layer that can extent up to several boundary layer thicknesses are often referred to as turbulent superstructures. These structures contain a relatively large portion of the layer’s turbulent kinetic energy and have been shown to interact with the near-wall flow structures. In the last few decades extensive research on zero-pressure gradient (ZPG) turbulent boundary layers has been done, however by comparison, the structural characteristics for adverse pressure gradient turbulent (APG) boundary layer flows are much less studied despite their strong significance aero-hydrodynamic vehicle design. Therefore, the three-dimensional dynamics of turbulent superstructures in a turbulent boundary layer flow are investigated in the Atmospheric Wind Tunnel Munich (AWM) using a multi-camera 3D time-resolved Lagrangian particle tracking approach. In this study, Lagrangian and Eulerian statistics will be used to characterize the dynamics and interaction of turbulent superstructures within a zero pressure gradient (ZPG) turbulent boundary layer at Reτ = 5000 or Reθ = 14 000 that then flows over a curved plate subjected to a favorable (FPG) and strong adverse (APG) pressure gradient, which eventually separates. An Eulerian analysis, using multi-point correlations of 3D velocity fields, found that the average superstructure topology is modulated by decelerating flow in the APG region when compared to the ZPG region, however the basic shape and spanwise pattern is preserved. Looking into the behavior of individual trajectories, it was found that the dispersion of single particles along trajectories in the log-law layer are capable of moving more than the average Eulerian superstructure spacing in the spanwise direction. Furthermore, the mean square of the single particle dispersion indicates that the maximum dispersion in the spanwise direction comes from particles released at the wall-normal location corresponding to the so-called “second-peak/plateau” region in the streamwise normal Reynolds stress.

Predicting flow resistance in open-channel flows with submerged vegetation

In vegetated flows, hydrodynamic parameters, such as drag coefficient, frontal area and deflected canopy height, influence velocity distributions, mean velocity and flow resistance. Previous studies have focused on flow–structure interaction in sparse vegetation, dense vegetation or transitional canopies, respectively. To date, a unifying approach to estimate hydrodynamic properties of submerged vegetated flows across the full vegetation density spectrum is missing. Herein, published data sets across a wide range of vegetation conditions were re-analysed using a previously proposed four-layer velocity superposition model. For the investigated vegetation conditions, the velocity model was able to match measured velocity distributions and depth-averaged mean velocity. The contribution of each velocity layer to the mean velocity was analyzed, showing that the mixing layer is dominant in transitional canopies with shallow submergence, and that the log-law layer is dominant in denser canopies with deeper submergence. Based upon velocity distributions, an explicit equation for the Darcy–Weisbach friction factors was deduced that is able to predict flow resistance as function of relative submergence. While each velocity distribution could be well described with the four-layer model across the range of vegetation conditions, some data scatter in model parameters was observed. To improve predictive capabilities of the model, future research should focus on detailed velocity measurements with high spatial resolution.

Wall-shear-stress measurements using volumetric µPTV

Accurately determining the wall-shear-stress, tau _textrm{w}, experimentally is challenging due to small spatial scales and large velocity gradients present in the near-wall region of turbulent flows. To avoid these resolution requirements, several indirect iterative fitting methods, most notably the Clauser chart method, exist for determining overline{tau }_{w} by fitting the mean velocity profile further away from the near-wall region in the log-law layer. These methods often require proper selection of fitting constants, assumptions of a canonical flow state, and other empirical-based generalizations. To reduce the amount of ambiguity, determining the near-wall velocity gradient by assuming a linear relationship between the mean streamwise velocity and wall normal distance in the viscous sublayer can be used. However, this requires an accurate unbiased measurement of the near-wall velocity profile in the region below five viscous spatial units, which can be less than 50 µm for high Reynolds number flows. Therefore, in this study a method for a volumetric defocusing microparticle tracking velocimetry method is presented that is capable of resolving the flow in the viscous sublayer of a turbulent boundary layer up to U_{textrm{e}}=44.7,m/s (Re_{theta }=27250). This method allows for the measurement of the near-wall flow through a single optical access for illumination and imaging and serves as an excellent complement of larger scale measurements that require near-wall information. The overline{tau }_textrm{w} values determined from the defocusing approach were found to be in good agreement values obtained from a simultaneous parallax PTV measurement. Furthermore, analysis of the diagnostic plot and cumulative distribution of measured fluctuations in the near-wall region, showed that both methods are capable of accurately determining mean velocity and fluctuation profiles in the self-similar viscous sublayer region.

Reinvestigating the Parabolic-Shaped Eddy Viscosity Profile for Free Surface Flows

The flow in rivers is turbulent. The main parameter related to turbulence in rivers is the eddy viscosity, which is used to model a turbulent flow and is involved in the determination of both velocities and sediment concentrations. A well-known and largely used vertical distribution of eddy viscosity in free surface flows (open channels and rivers) is given by the parabolic profile that is based on the logarithmic velocity profile assumption and is valid therefore only in the log-law layer. It was improved thanks to the log-wake law velocity profile. These two eddy viscosities are obtained from velocity profiles, and the main shortcoming of the log-wake profile is the empirical Coles’ parameter. A more rigorous and reliable analytical eddy viscosity model is needed. In this study, we present two analytical eddy viscosity models based on the concepts of velocity and length scales, which are related to the exponentially decreasing turbulent kinetic energy (TKE) function and mixing length, namely, (1) the exponential-type profile of eddy viscosity and (2) an eddy viscosity based on an extension of von Karman’s similarity hypothesis. The eddy viscosity from the second model is Re*-independent, while the eddy viscosity from the first model is Re*-dependent (where Re* is the friction Reynolds number). The proposed analytical models were validated through computation of velocity profiles, obtained from the resolution of the momentum equation and comparisons to experimental data. With an additional correction function related to the damping effect of turbulence near the free surface, both models are similar to the log-wake-modified eddy viscosity profile but with different values of the Coles’ parameter, i.e., Π=0.2 for the first model and Π=0.15 for the second model. These values are similar to those found in open-channel flow experiments. This provides an explanation about the accuracy of these two analytical models in the outer part of free surface flows. For large values of Re* (Re* > 2000), the first model becomes Re* independent, and the two coefficients reach asymptotic values. Finally, the two proposed eddy viscosity models are validated by experimental data of eddy viscosity.

Assessment of the wall-adapting local eddy-viscosity model in transitional boundary layer

The main idea of this manuscript is to assess the wall-adapting local eddy-viscosity (WALE) model, which is designed for large-eddy simulation (LES) of turbulent boundary layer, in transitional flow. In contrast to many other sub-grid-scale (SGS) models, the WALE model demonstrates the asymptotic decay of the eddy viscosity in the vicinity of a solid wall in turbulent boundary layer without relying on a dynamic procedure on the SGS model coefficient. Furthermore, the WALE model yields zero eddy viscosity in pure shear flow. Such features are attractive for LES of laminar-to-turbulent transition, yet the WALE model has not been thoroughly investigated in transitional boundary layer flow. Well-resolved LES is conducted for canonical boundary layer transition triggered by sub-harmonic resonance. The model formulation is thoroughly analyzed in the transition process. The asymptotic behavior of the two major tensors, i.e., the strain rate tensor and the traceless tensor of the velocity gradient squared, are confirmed in the current well-resolved LES. The cubic decay of the eddy viscosity in the wall distance is confirmed from the asymptotic analysis on the transitional flow. The presence of the strain rate in the model formulation generates practically zero eddy viscosity in the pre-transition region, allowing interactions of small-amplitude instabilities. The traceless tensor of the velocity gradient squared escalates if small-scale eddies appear, leading to sizable eddy viscosity particularly in the log-law layer in turbulent flow. The response of the WALE model to grid resolution is also discussed. The performance of the WALE model is compared to two other SGS models, Smagorinsky and Vreman models with a constant coefficient, in the current transitional wall-bounded flow.

Vortex dynamics in low- and high-extent polymer drag reduction regimes revealed by vortex tracking and conformation analysis

Turbulent flow profiles are known to change between low- (LDR) and high-extent drag reduction (HDR) regimes. It is however not until recently that the LDR-HDR transition is recognized as a fundamental change between two DR mechanisms. Although the onset of DR, which initiates the LDR stage, is explainable by a general argument of polymers suppressing vortices, the occurrence of HDR where flow statistics are qualitatively different and DR effects are observed across a much broader range of wall regions remains unexplained. Recent development of the vortex axis tracking by iterative propagation algorithm allows the detection and extraction of vortex axis-lines with various orientations and curvatures. This new tool is used in this study to analyze the vortex conformation and dynamics across the LDR-HDR transition. Polymer effects are shown to concentrate on vortices that are partially or completely attached to the wall. At LDR, this effect is an across-the-board weakening of vortices which lowers their intensity without shifting their distribution patterns. At HDR, polymers start to suppress the lift-up of streamwise vortices in the buffer layer and prevent their downstream heads from rising into the log-law layer and forming hairpins and other curved vortices. This interrupts the turbulent momentum transfer between the buffer and log-law layers, which offers a clear pathway for explaining the distinct mean flow profiles at HDR. The study depicts the first clear physical picture regarding the changing vortex dynamics between LDR and HDR, which is based on direct evidence from objective statistical analysis of vortex conformation and distribution.

Non-uniform flow over cobble bed with submerged vegetation strip

To better understand the influence of non-uniform flow in coarse-bed streams with a submerged vegetation strip on flow characteristics, data were collected from experiments in a laboratory flume and field investigations in a cobble-bed river reach in central Iran. The results of the experiments and field observations showed that the velocity profiles for non-uniform flows in cobble-bed streams with vegetation strips can be divided into either two or three sub-zones. The velocity profiles of non-uniform flows in the laboratory flume and cobble-bed stream deviated from the log law in the wake and mixing zones, but fitted approximately in the log-law layer zone. The distribution of Reynolds stress for non-uniform flows was influenced by the vegetation strip and showed a non-concave shape. Near the channel bed, sweep motion occurred more frequently than the ejection process. Close to the top of vegetation strip, the correlation coefficient (ru′w′) was negative, indicating downward momentum transport due to sweep and ejection processes. However, ru′w′was positive near the water surface, indicating upward momentum by the vegetation strip due to inward and outward interactions. The study shows that laboratory results cannot be easily applied to natural rivers without considerable assumptions.

Eddy Viscosity and Velocity Profiles in Fully-Developed Turbulent Channel Flows

In this study, we present eddy viscosity formulations based on analytical solutions for turbulent kinetic energy (TKE) in fully-developed turbulent channel flows. Mean streamwise velocities are obtained by solving the momentum equation with the proposed analytical eddy viscosity formulations. Our results were validated by comparisons with DNS data of fully-developed turbulent channel flows for 300 < Reτ < 2000 (where Reτ denotes the friction Reynolds number defined by the friction velocity uτ, kinematic viscosity ν and the channel half-width δ). Our method allows accurate description of mean streamwise velocity profiles and turbulent shear stress outside the log-law layer, in the viscous and buffer layers and in the outer region. The aim of this study is to provide simple analytical tools for turbulent channel flow applications.

Large-Eddy Simulation Study of Log Laws in a Neutral Ekman Boundary Layer

Abstract The characteristics of wind profiles in a neutral atmospheric boundary layer and their dependence on the geostrophic wind speed Ug, Coriolis parameter f, and surface roughness length z0 are examined utilizing large-eddy simulations. These simulations produce a constant momentum flux layer and a log-law layer above the surface characterized by a logarithmic increase of wind speed with height. The von Kármán constant derived from the mean wind profile is around 0.4 over a wide range of control parameters. The depths of the simulated boundary layer, constant-flux layer, and surface log-law layer tend to increase with the wind speed and decrease with an increasing Coriolis parameter. Immediately above the surface log-law layer, a second log-law layer has been identified from these simulations. The depth of this upper log-law layer is comparable to its counterpart in the surface layer, and the wind speed can be scaled as , as opposed to just in the surface log-law layer, implying that in addition to surface processes, the upper log-law layer is also influenced by Earth’s rotation and large-scale conditions. Here is the friction velocity at the surface, and h is the boundary layer depth. An analytical model is proposed to assist in the interpretation of the log laws in a typical Ekman boundary layer. The physics and implications of the upper log-law layer are discussed.

Distinct transition in flow statistics and vortex dynamics between low- and high-extent turbulent drag reduction in polymer fluids

Flexible polymer additives are known to reduce the energy dissipation and friction drag in turbulent flows. As the fluid elasticity increases, the flow undergoes several stages of transitions. Much attention in the area has been focused on the onset of drag reduction (DR) and the eventual convergence to the maximum drag reduction (MDR) asymptote. Between the onset and MDR, recent experimental and numerical observations prompted the need to further distinguish the low- and high-extent drag reduction (LDR and HDR). Fundamental knowledge of this transition will be important for understanding turbulent dynamics in the presence of polymers, as well as for inspiring new flow control strategies for efficient friction reduction. We use direct numerical simulation (DNS) to explore all these transitions in the parameter space and, in particular, demonstrate that the LDR–HDR transition is not merely a quantitative effect of the level of drag reduction, but a qualitative transition into a different stage of turbulence. A number of sharp changes in flow statistics are found to accompany the transition and at HDR, turbulence becomes localized with vortices forming clusters. These observations suggest that polymer-induced drag reduction follows two distinct stages. The first starts at the onset of drag reduction, where the coil-stretch transition of polymers causes an overall suppression of turbulent fluctuations. The second starts at the LDR–HDR transition, where flow statistics become fundamentally changed in the log-law layer and turbulence localization is observed. A mechanism is then proposed for the latter based on the changing vortex regeneration dynamics between LDR and HDR.

On the validity of the quasi-steady-turbulence hypothesis in representing the effects of large scales on small scales in boundary layers

The “quasi-steady hypothesis,” as understood in the context of large-scale/small-scale interactions in near-wall turbulence, rests on the assumption that the small scales near the wall react within very short time scales to changes imposed on them by energetic large scales whose length scales differ by at least one order of magnitude and whose energy reaches a maximum in the middle to the outer portion of the log-law layer. A key statistical manifestation of this assumption is that scaling the small-scale motions with the large-scale wall-friction-velocity footprints renders the small-scale statistics universal. This hypothesis is examined here by reference to direct numerical simulation (DNS) data for channel flow at Reτ ≈ 4200, subjected to a large-scale/small-scale separation by the empirical mode decomposition method. Flow properties examined include the mean velocity, second moments, joint probability density functions, and skewness. It is shown that the validity of the hypothesis depends on the particular property being considered and on the range of length scales of structures included within the large-scale spectrum. The quasi-steady hypothesis is found to be well justified for the mean velocity and streamwise energy of the small scales up to y+∼O(80), but only up to y+∼O(30) for other properties.

Numerical analysis of flow around porous rectangular cylinders with uniform injection or suction

The flow around rectangular cylinders of various aspect ratios of B/H = 2, 3 and 4 with uniform injection or suction through the rear and top/bottom surfaces has been studied numerically at a Reynolds number Re = 21,400. The effects of injection or suction through the cylinder surfaces have been linked to the analysis via the wall functions of the viscous sub–layer and log–law layer. The injection or suction has been characterised by an injection coefficient CQ = vw/u∞, which takes negative values for suction. The governing equations, which are a combination of the ensemble–averaged Navier–Stokes equations for incompressible flow in conjunction with the equations of a modified k–ε turbulence model, have been solved by the finite volume method. A non–uniform staggered grid arrangement has been used to economise on computing time. The results show that the vortex structure in the wake is influenced primarily by suction through the top and bottom surfaces of the rectangular cylinders. In addition, the parameters taken into account such as drag coefficient and Strouhal number are also affected by the injection coefficient as well as the injection/suction surfaces of the rectangular cylinders.

A universal scaling for low-order structure functions in the log-law region of smooth- and rough-wall boundary layers

AbstractWe consider the log-law layer of both smooth- and rough-wall boundary layers at large Reynolds number. A scaling theory is proposed for low-order structure functions (say$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n \leq 6$) in the range of scales$\eta \ll r \ll \delta $, where$\eta $is the Kolmogorov length and$\delta $is the boundary layer thickness. This theory rests on the hypothesis that the turbulence in this intermediate range of scales depends only on the scale$r$, the local dissipation rate and the shear velocity. Crucially, the structure of the turbulence is assumed to be independent of the distance from the wall,$y$, except to the extent that$y$sets the value of the local dissipation rate. A detailed comparison is made between the predictions of the theory and data taken from both smooth- and rough-wall boundary layers. The data support the hypothesis that it is the dissipation rate, and not$y$, that controls the structure of the turbulence for this range of eddy sizes. Our findings provide the first unified scaling theory for both smooth- and rough-wall turbulence.

Investigation on the wall function implementation for the prediction of ship resistance

A computational fluid dynamics (CFD) code, dubbed SNUFOAM, was developed to predict the performance of ship resistance using a CFD tool kit with open source libraries. SNUFOAM is based on a pressure-based cell-centered finite volume method and includes a turbulence model with wall functions. The mesh sensitivity, such as the skewness and aspect ratio, was evaluated for the convergence. Two wall functions were tested to solve the turbulent flow around a ship, and the one without the assumption of the equilibrium state between turbulent production and dissipation in the log law layer was selected. The turbulent flow around a ship simulated using SNUFOAM was compared to that by a commercial CFD code, FLUENT. SNUFOAM showed the nearly same results as FLUENT and proved to be an alternative to commercial CFD codes for the prediction of ship resistance performance.

Improved finite volume discretisations of entropy generation and turbulence production

The finite volume discretisation of the rates of turbulence production and entropy generation commonly use cell centred differencing schemes. Whilst these are easy to derive they can result in discrepancies in solutions that are otherwise based on gradients approximated at cell faces. For example, a turbulent log law layer calculated using the k– ϵ model can exhibit a spurious peak in the turbulence kinetic energy near the wall, or the volumetric rate of work calculated from entropy generation may not equal energy fluxes through domain boundaries. In this article new discretisations of turbulence production and entropy generation are derived that are based on derivatives centred at cell faces. These do not display the erroneous behaviours of the cell centre discretisations.

Effects of viscoelasticity on the probability density functions in turbulent channel flow

The probability density functions (PDFs) of the velocity fluctuations and their derivatives of a viscoelastic turbulent channel flow are calculated and compared against those for a Newtonian fluid at a friction Reynolds number 180. The velocity fields in both cases are provided from previous and new direct numerical simulations. In the viscoelastic case, the Giesekus model is used at a friction Weissenberg number 50, a mobility parameter 1/900, and viscosity ratio 0.9, corresponding to 37.4% drag reduction. The skewness and the flatness factors, which are also calculated and presented as functions of the distance from the wall, further reveal and quantify the non-Gaussian characteristics of the turbulent structures and how they are distributed in the flow domain. With the presence of viscoelasticity the non-Gaussian character of the PDFs is typically further enhanced. In particular, larger asymmetries and much longer tails are typically observed in the viscoelastic PDFs. This indicates higher intermittency in the viscoelastic turbulent flow versus the Newtonian one, along with strong correlations between all scales of turbulence. Furthermore, we have also seen significant qualitative changes regarding the distribution of the non-Gaussian characteristics of the PDFs as a function of the distance from the wall. The most notable differences are seen in the PDFs of the shear and spanwise velocity components in the buffer and log-law regions, as well as in most of the components of the first and second order velocity derivatives. It is also found that in the log-law layer the PDFs of the logarithm of the dissipation and the pseudodissipation rate, as well as of the enstrophy, are almost Gaussian. In contrast, in the viscous sublayer and the buffer layer, significant deviations from the Gaussian shape are observed. Whereas the deviations from Gaussianity are more enhanced with viscoelasticity next to the wall, with the PDFs even more negatively skewed, away from the wall sometimes (i.e., for the logarithm of the enstrophy) they are decreased. All these findings are pointing out to a fairly complex picture for the interaction of viscoelasticity with turbulence that makes the task of developing turbulent viscoelastic models even more challenging.

Numerical Simulation of Turbulent Flow Around a Square Cylinder with Uniform Injection or Suction and Heat Transfer

In the present study, the turbulent flow around a square cylinder with uniform injection or suction through various surfaces has been numerically studied. The wall functions of the viscous sublayer and log-law layer for velocity and temperature on the cylinder have been modified to allow for the effect of the injection or suction through the cylinder. The finite-volume method has been used for solving the ensemble averaged Navier-Stokes equations for incompressible flow in conjunction with the k − ϵ turbulence model equations that include the Kato-Launder's modification. A nonuniform staggered grid arrangement has been used. The results show that the parameters taken into account such as drag coefficient, and Strouhal and Nusselt numbers have been influenced by the uniform injection and suction through certain surfaces of the square cylinder, as well as the flow field parameters.