The traditional laminar plane wall jet is studied when the medium is filled with nanoparticles of Ag, Cu, CuO, Al O and TiO . It is aimed to understand the effects of several nanofluids on the heat and flow behaviors of the wall jet. Momentum and thermal integral flux relations are obtained initially. Later on, some important shape factors are defined designing the momentum boundary layer, shear layer as well as the thermal boundary layer when the wall is subjected to either adiabatic or isothermal wall constraints. By means of these parameters, the flow field is shown to be decelerated and as a consequence the shear stress on the wall is enhanced. Without solving the energy equation, the thermal layer shape factor enables one to fully seize the cooling effect of considered nanofluids for both adiabatic and isothermal wall cases. As a result, the heat transfer rate is found to be greatly enhanced by the presence of nanoparticles. Same conclusions are reached by two different popular nanofluid models made use in the recent nanofluid researches.
In recent years Discontinuous Galerkin (DG) methods have emerged as one of the most promising high-order discretization techniques for CFD. DG methods have been successfully applied to the simulation of turbulent flows by solving the Reynolds averaged Navier–Stokes (RANS) equations with first-moment closures. More recently, due to their favorable dispersion and dissipation properties, DG discretizations have also been found very well suited for the Direct Numerical Simulation (DNS) and Implicit Large Eddy Simulation (ILES) of turbulent flows. The growing interest in the implementation of DG methods for DNS and ILES is motivated by their attractive features. In particular, these methods can easily achieve high-order accuracy on arbitrarily shaped elements and are perfectly suited to -adaptation techniques. Moreover, their compact stencil is independent of the degree of polynomial approximation and is thus well suited for implicit time discretization and for massively parallel implementations. In this paper we focus on recent developments and applications of an implicit high-order DG method for the DNS and ILES of both compressible and incompressible flows. High-order spatial and temporal accuracy has been achieved using the same numerical technology in both cases. Numerical inviscid flux formulations are based on the exact solution of Riemann problems (suitably perturbed in the incompressible case), and viscous flux discretizations rely on the BR2 scheme. Several types of high-order (up to order six) implicit schemes, suited also for DAEs, can be employed for accurate time integration. In particular, linearly implicit Rosenbrock-type Runge–Kutta schemes have been used for all the simulations presented in this work. The massively separated incompressible flow past a sphere at , with transition to turbulence in the wake region, is considered as a DNS test case, while the potential of the ILES is demonstrated by computing the compressible transitional flow at , and , around the Selig–Donovan 7003 airfoil. The computed solutions are compared with experimental data and numerical results available in the literature, showing good agreement.
Linear stability analysis has proven to be a useful tool in the analysis of dominant coherent structures, such as the von Kármán vortex street and the global spiral mode associated with the vortex breakdown of swirling jets. In recent years, linear stability analysis has been applied successfully to turbulent time-mean flows, instead of laminar base-flows, which requires turbulent models that account for the interaction of the turbulent field with the coherent structures. To retain the stability equations of laminar flows, the Boussinesq approximation with a spatially nonuniform but isotropic eddy viscosity is typically employed. In this work we assess the applicability of this concept to turbulent strongly swirling jets, a class of flows that is particularly unsuited for isotropic eddy viscosity models. Indeed we find that unsteady RANS simulations only match with experiments with a Reynolds stress model that accounts for an anisotropic eddy viscosity. However, linear stability analysis of the mean flow is shown to accurately predict the global mode growth rate and frequency if the employed isotropic eddy viscosity represents a least-squares approximation of the anisotropic eddy viscosity. Viscosities derived from the model did not achieve a good prediction of the mean flow nor did they allow for accurate stability calculations. We conclude from this study that linear stability analysis can be accurate for flows with strongly anisotropic turbulent viscosity and the capability of the Boussinesq approximation in terms of URANS-based mean flow prediction is not a prerequisite.
To gain insight into characteristic wake flow modes, which among others are responsible for asymmetrical loads on the engine extensions of space launchers at transonic speeds, combined experimental and numerical investigations of a turbulent wake flow are performed at and . The experiments are conducted at the Bundeswehr University Munich using planar PIV and wall pressure measurements, while the numerical investigations are performed by the Institute of Aerodynamics at RWTH Aachen University using a zonal RANS–LES approach and dynamic mode decomposition (DMD). The analysis is done on a planar space launcher configuration that consists of a backward-facing step with a long shock-free forebody avoiding undesired shock-boundary-layer interactions upstream of the analyzed wake flow region. The investigated wake flow is characterized by a highly unsteady behavior of the shear layer shedding from the forebody and subsequently reattaching onto the splitter plate. The strong variation of the reattachment positions in the spanwise and the streamwise direction leads to pronounced wall pressure oscillations and consequently, structural loads. By means of a classical statistical analysis, i.e., spatial and temporal Fourier transforms and two-point correlations of experimental and numerical data, one spatial coherent scale with a spanwise wavelength of and two characteristic frequencies of and have been obtained for the investigated wake flow problem. To clarify the coherent fluid motion dominating the detected spatio-temporal behavior, a DMD algorithm is applied to the time-resolved three-dimensional streamwise velocity field. The frequencies of the first two extracted stable sparsity DMD modes closely coincide with the characteristic peaks in the wall pressure spectra. The analysis of the three-dimensional shape of the extracted DMD modes reveals that the detected wake flow behavior is caused by a pronounced low-frequency cross-pumping motion of the recirculation region and a high-frequency cross-flapping motion of the shear layer, respectively. Both wake flow modes feature a pronounced nearly-periodical variation in the spanwise direction with an approximate spanwise wavelength of two step heights. Thus, the extracted underlying wake modes clearly explain the occurrence of corresponding peaks in the wall pressure spectra, the periodical formation of wedge-shaped reattachment regions on the splitter plate, and the detection of finger-like structures in the PIV experiments.
The present paper provides an experimental optimization of a NACA 4415 airfoil equipped with vortex generators (VGs) to control its flow separation. To build this optimal configuration an experimental parametric study was conducted on five geometrical parameters: thickness and height of vortex generators, position, orientation angle with respect to the mean flow direction, spacing in the spanwise direction. Moreover, a new configuration that includes micro generators behind the conventional ones was also investigated as a potentially interesting solution. For all these cases wind tunnel tests were performed and compared for different angles of attack and various Reynolds numbers up to 2 . These experiments enabled us to highlight the main trends to get an optimal design, for which quantitative improvement can be achieved by passive means in terms of aerodynamic performances on NACA4415 airfoil. The results reveal that triangular shape vortex generators are best suited to control boundary layer separation. An optimum angle of VGs is obtained for 12°with a 3 mm distance between vortex generators located at 50% of the chord. It was found that micro vortex generators are very effective in controlling the flow with less parasite drag. The maximum lift coefficient for an airfoil with coupled vortex generators increases by 21% and a flow separation is delayed by 17°. However, this very good performance is counterbalanced by the appearance of parasitic drag. Indeed, it creates a counter-rotating array of vortices with the second raw of micro-vortex generators that reinforce the vortexes strength without any increase in device height.
Wave-interference effects on the far-field waves created by a catamaran, with identical twin hulls of length at a lateral separation distance , that advances at constant speed along a straight path in calm water of large depth are considered. Systematic computations are performed for 125 Froude numbers (where denotes the acceleration of gravity) within the range , 25 hull spacings within the range , and seven simple mathematically-defined hulls that correspond to a broad range of main hull-shape parameters (beam/length, draft/length, beam/draft, waterline entrance angle), i.e. for 21,875 distinct cases in all. The two dominant wake angles and that correspond to the ray angles and where the largest divergent waves are found within the Kelvin wake with are determined numerically via a realistic yet practical method. This practical method, used previously for monohull ships, is based on the numerical determination of the two major peaks of the amplitude function in the Fourier-Kochin representation of far-field ship waves, evaluated via the Hogner approximation and the stationary-phase approximation. The computations show that the hull shape only has a relatively small influence on the wake angles and associated with the dominant waves. A useful practical consequence of this numerical finding is that the two wake angles and can be estimated, without computations, for general catamarans in terms of the Froude number and the hull spacing via simple analytical approximations, obtained here via parametric computations. The computations also show that lateral interference effects are dominant if and/or are sufficiently large, i.e. for ‘wide’ and/or ‘fast’ catamarans. Moreover, the computations reported here for catamarans represented via hull-surface distributions of sources show that the basic two-point wavemaker model of interference between the waves created by the twin bows of a catamaran is realistic for wide and/or fast catamarans. However, wave-interference effects are more complicated if both and are small, i.e. for ‘narrow slow’ catamarans. The numerical analyses of dominant waves considered earlier for monohull ships and here for catamarans show that, although the of the waves created by a ship are strongly influenced by the shape of the ship hull, the where the largest waves are found are mostly a feature that is only weakly influenced by the hull shape and the related wave amplitude.
In this analysis, the flow and heat transfer characteristics of a nanofluid over a stretching/shrinking surface with suction are investigated. Using a similarity transformation, the nonlinear system of partial differential equations is converted into nonlinear ordinary differential equations. These resulting equations are solved analytically and numerically using a collocation method. Multiple (dual: upper and lower branch) solutions are shown to exist in a range of the governing parameters. In addition, the reduced skin friction coefficient and the reduced heat transfer from the surface of the sheet as well as the velocity, temperature and concentration profiles are analyzed subject to several parameters of interest, namely suction parameter, Brownian motion and thermophoresis parameters, Prandtl number, nanofluid Lewis number and dimensionless slip parameter. The results indicate that the skin friction coefficient and the heat transfer from the surface of the sheet increase with suction effect. It is also observed that suction widens the range of the stretching/shrinking parameter for which the solution exists.
A unified formulation for stagnation-point flows and linearly stretching plates is given wherein the two can occur separately or in unison. Reductions to known cases are given. It is noticed that previous work on stretching plates beneath planar and axisymmetric stagnation point flows have respectively aligned planar stretching and axisymmetric stretching. The general formulation reveals other combinations of stretching beneath stagnation-point flows exist and three new cases are studied in detail. The linear stability of dual and multiple solutions are calculated.
Unsteady cavitating turbulent flow around a twisted wing is simulated by using the large eddy simulation method. Two sets of grids with 10 million and 2 million nodes are used to investigate the influence of mesh resolution on the results. The results of non-cavitating flow with the coarse mesh agree well with the experiment, but the accuracy of the fine mesh results is remarkably higher for the cavitating flow, which indicates that more nodes are needed for the cavitating flow simulation than the non-cavitating flow. Then the parameters affecting the grid resolution are investigated. It is observed that the small size shedding vortex can only be captured by the fine mesh and the smaller grid spacing in the spanwise direction is needed to capture the details of re-entry jet. The re-entry jet in spanwise direction can affect the overall development of cavities, which is sensitive to the pressure gradient and spanwise resolution of the mesh.
The objective of this work is to investigate numerically the different physical mechanisms of the transition to turbulence of a separated boundary-layer flow over an airfoil at low angle of attack. In this study, the spectral elements code 5000 is used to simulate the flow over a SD7003 wing section at an angle of attack of . Several laminar cases are first studied from to , and a gradual increase of the Reynolds number is then performed in order to investigate one transitional case at . Computations are compared with measurements where the instability mechanisms in the separated zone and near wake zone have been analyzed. The mechanism of transition is investigated, where the DMD (Dynamic Mode Decomposition) is used in order to extract the main physical modes of the flow and to highlight the interaction between the transition and the wake flow. The results suggest that the transition process appears to be physically independent of the wake flow, while the LSB shedding process is locked-in with the von Kármán instability and acts as a sub-harmonic.
Existing differences between experimental, computational and theoretical representations of a particular flow do not allow one-to-one comparisons, prevent us from identifying the absolute contributions of the various sources of uncertainty in each approach, and highlight the importance of developing suitable corrections for experimental techniques. In this study we utilize the latest Pitot tube correction schemes to develop a technique which improves on the outcome of hot-wire measurements of mean velocity profiles in ZPG turbulent boundary layers over the range . Measurements by Bailey et al. (2013), carried out with probes of diameters ranging from 0.2 to 1.89 mm, supplemented by other data with larger diameters up to 12.82 mm, are used first to develop a somewhat improved Pitot tube correction which is based on viscous, shear and near-wall schemes (which contribute with around of the effect), together with a turbulence scheme which accounts for of the whole correction. The correction proposed here leads to similar agreement with available high-quality datasets in the same Reynolds number range as the one proposed by Bailey et al. (2013), but this is the first time that the contribution of the turbulence scheme is quantified. In addition, four available algorithms to correct wall position in hot-wire measurements are tested, using as benchmark the corrected Pitot tube profiles with artificially simulated probe shifts and blockage effects. We find that the -Musker correction developed in this study produces the lowest deviations with respect to the introduced shifts. Unlike other schemes, which are based on a prescribed near-wall region profile description, the -Musker is focused on minimizing the deviation with respect to the relation, characteristic of wall-bounded turbulent flows. This general approach is able to locate the wall position in probe measurements of the wall-layer profiles with around one half the error of the other available methods. The difficulties encountered during the development of adequate corrections for high- boundary layer measurements highlight the existing gap between the conditions that can be reproduced and measured in the laboratory and the so-called canonical flows.
An analysis has been carried out to obtain the flow, heat and mass transfer characteristics of a viscous fluid having temperature dependent viscosity and thermal conductivity in a vertical channel. The energy equation accounts for viscous dissipation, while the first order homogeneous chemical reaction between the fluid and diffusing species is included in the mass diffusion equation. The walls of the channel are maintained at constant but different temperatures. The non-dimensional coupled nonlinear ordinary differential equations are solved analytically using perturbation method and numerically using Runge–Kutta shooting method. The velocity, temperature and concentration distributions are obtained numerically and presented through graphs. Skin friction coefficient and Nusselt number at the walls of the channel are derived and discussed and their numerical values for various values of physical parameters are presented through tables.
In the present paper two numerical schemes for propagating waves over a variable bathymetry in an existing High-Order Spectral (HOS) model are introduced. The first scheme was first developed by Liu and Yue (1998), and the second one is an improved scheme which considers two independent orders of non-linearity: one for the bottom and one for the free-surface elevation. We investigate the numerical properties (accuracy, convergence rate, efficiency) of both schemes with respect to the numerical parameters on a simple configuration. To validate the proposed schemes, we first consider Bragg reflection from a sinusoidal bottom patch — as an example of a small bottom variation around a mean water depth. The second validation case focuses on a larger bottom variation with the study of the shoaling of linear waves. Finally, an application is performed to demonstrate the applicability of the proposed model to non-linear cases where the bottom variation is important. In this concern, the very well documented test case of a 2D underwater bar is studied in detail. Comparisons are provided with both experimental and numerical results.
The interaction of closely spaced bubbles is relevant for a variety of geophysical, industrial, and medical applications. The present study fills a research gap in that it is concerned theoretically with the volume pulsations of bubbles at small distances compared with their sizes. It was shown that the bi-spherical coordinates provide separation of variables and are more suitable for analysis of this problem. A coupled oscillator method is used to describe collective acoustic resonances from closely spaced bubbles in water. Explicit formulas have been derived that describe the dependence of the bubbles natural frequencies and damping coefficients on their sizes and the separation distance.
Errors that stem from a practical analytical approximation to the local flow component in the Green function associated with steady linear potential flow around a ship hull are considered. Although the approximation is not very accurate, the flow potentials evaluated via the exact local flow component or the approximation for Froude numbers , 0.3 and 0.5 cannot be distinguished, except at for which relatively small differences can be observed. Moreover, the sinkage, the trim angle and the wave drag predicted by the Neumann–Michell (NM) theory, with the local flow potential evaluated using or , are in very close agreement. Despite its remarkably simplicity, the analytical approximation can then be used to compute the local flow component in the Green function, in the entire flow region, within the framework of the NM linear potential flow theory and the related Hogner approximation. The practical analytical approximation is far more practical than the basic integral representation of , and is an important element of the NM theory. Indeed, the analytical approximation , and other features of the NM theory, make it possible to compute the flow around a steadily advancing ship hull in a highly efficient way, as required for routine practical applications to design and hull-form optimization.
Pressurization and cryogenic conditions have been used in some experiments to change the kinematic viscosity of the flowing gas by many orders of magnitude in order to achieve high Reynolds number conditions in facilities of limited size. This leads to a substantial reduction of the viscous length scale , as in the so-called Princeton “Superpipe” experiments. We demonstrate that the limited dimensions of the facilities and probes can lead to inaccuracies in the near-wall measurements for increasing Reynolds number. Specifically, a lack of accurate wall-normal probe positioning is simulated using three different datasets of wall-bounded turbulent flows. Relatively large errors in the overlap region parameters are observed for position errors of small physical magnitude that become greatly amplified in wall units as is reduced. This offers an alternative interpretation to some of the key findings reported by the Superpipe team, such as the increasing lower limit of the logarithmic region , the existence of a power law region between the wall and the logarithmic layer, and the “mixing transition” phenomenon in wall-bounded turbulence.
Gas–liquid density ratio ( ) is a key dimensionless number in sloshing assessment methodologies of membrane containment systems for LNG tanks of floating structures. Earlier studies on the effect of were mainly statistical and effects of were usually mixed with those of gas compressibility and ullage gas pressure but attributed only to . In an attempt to separately study such effects, part I of this work studied the effects of far from impact zones (global effects of gas–liquid density ratio) which proved to be small in the studied range of (0.0002 to 0.0060). The effects of near impact zones and in the instants prior to the detection of any compressibility effects are referred to as local effects and are treated in the current paper (part II). The test setup was identical to the one presented in Part I and consisted of two 2D model tanks representing transverse slices of tank 2 (out of 4) of a membrane LNG carrier with total capacity of 152000 m at scales 1:20 and 1:40. Both model tests were performed at 20% fill level of the tank heights. Water was the main liquid that was used. In some tests at scale 1:20 a solution of sodium polytungstate (SPT) was also used which had a higher density compared to water. Different ullage gases of helium (He), air, two mixtures of sulfur hexafluoride (SF ) and nitrogen (N ), and pure SF , all at atmospheric pressure with a range of s from 0.0002 to 0.0060 were utilized. Synchronized high-speed video cameras (@4000 fps) and arrays of piezo-electric PCB (112A21 and 112M361) pressure sensors (@40 kHz) monitored and measured impacts on the tank walls. In Part II of the study short and more regular tank motions which generated highly repeatable single impact waves (SIW) were used instead of long irregular tank motions which were considered in part I. By comparing the single impact waves (SIW) generated by identical tank motions but with different s, it was observed that clearly modifies wave shapes prior to the moment of wave breaking. Larger s tend to slow down the wave front and delay breaking. It was also observed that larger s slightly slow down wave trough runup as well. Those effects would also lead to a mild shift of impact types by changing the (for example Flip-through to slosh or large gas-pocket to small gas-pocket impacts). By comparing single impact waves (SIW) generated by identical tank motions and the same but with different gas and liquid densities it was shown that keeping the same is essentially needed to keep the same impact geometry as recommended by the existing sloshing assessment methodologies. Free surface instabilities were also very similar for those waves generated with the same tank motions and similar but with different gases and liquids. Considering the reduction of wave kinetic energy by heavier ullage gases as a relevant source of the statistical reduction of impact pressures and having in mind the mild shift of wave impact types caused by the change of it is still to be studied further why the heavier gas leads to smaller statistical pressures.
In the study of surface waves in the presence of a shear current, a useful and much studied model is that in which the shear flow has constant vorticity. Recently it was shown by Constantin (2011) that a flow of constant vorticity can only permit waves travelling exactly upstream or downstream, but not at oblique angles to the current, and several proofs to the same effect have appeared thereafter. Physical waves cannot possibly adhere to such a restriction, however. We resolve the paradox by showing that an oblique plane wave propagating atop a current of constant vorticity according to the linearised Euler equation carries with it an undulating perturbation of the vorticity field, hence is not prohibited by the Constantin theorem since vorticity is not constant. The perturbation of the vorticity field is readily interpreted in a Lagrangian perspective as the wave motion gently shifting and twisting the vortex lines as the wave passes. In the special case of upstream or downstream propagation, the wave advection of vortex lines does not affect the Eulerian vorticity field, in accordance with the theorem. We conclude that the study of oblique waves on shear currents requires a formalism allowing undulating perturbations of the vorticity field, and the constant vorticity model is helpful only in certain 2D systems.
We have developed a strong-coupling approach based on a uniformly-applied Eulerian description for both fluid and solid and provided a simple monolithic formulation to compute highly flexible structures interacting with surrounding fluid flows. Using a fast-tracking method and a fast solver for the modified pressure equation with variable density, we keep the same low computational cost as in the uniform density case studied previously. The new algorithm is first validated by the simulation of the self-sustained oscillation of a flexible plate. Then, it is applied to study the effect from density ratio on a flexible plate flapping with incoming flow. The simulation shows strong effect of density ratio on the pattern of fluid–structure interaction and the propulsion performance through the change in mass ratio and frequency ratio.
The mathematical model describing the counter-current imbibition phenomenon in a heterogeneous porous medium gives rise to a non-linear partial differential equation. This equation has been solved using homotopy analysis methods together with appropriate boundary and initial conditions. The solution represents the saturation of injected water during counter-current imbibition in a secondary oil recovery process, when water is injected. This saturation of injected water increases as the distance, , from the imbibition phase increases, for a given time . It is also deduced that the saturation of injected water in a homogeneous porous matrix is higher than the saturation of injected water in a heterogeneous porous matrix for the same distance and time . The numerical and graphical presentation of the saturation of injected water in heterogeneous as well as homogeneous porous matrices for distance and time are obtained in Maple.