Non-dimensional Pressure Gradient Research Articles

Bubble dynamics in a pressure gradient with reentrant jet break through and energy loss

The dynamics of a bubble in a pressure gradient is investigated experimentally and numerically with particular emphasis on the behavior at reentrant jet impact and break through the opposite side of the bubble with corresponding energy loss and vorticity generation. High speed photography observations of a bubble generated by electric spark energy deposit in a low ambient pressure tank are coupled with wavelet based Optical Flow Velocimetry (wOFV) and Boundary Element Method (BEM) numerical analysis to examine the flow field resulting from the reentrant jet formation and break through. We study, as an illustration, the effects of the constant pressure gradient due to gravity on the bubble dynamics. Energy losses between the first and second cycle are measured for the bubbles generated under various conditions characterized by a non-dimensional pressure gradient parameter, and the corresponding measured energy loss is used in the numerical simulations. Good correspondence is seen between the image analysis, the wOFV computations, and the BEM results and insight is gained on the involved physics.

Numerical analysis of two-layered isothermal calendering of viscoplastic and Newtonian fluids with different viscosity ratios

Co-extruded multi-layer plastic sheets and polymer structures formed by calendering process or by cold rolling are widely used in the packaging industry and thin-film transistor manufacturing. The different materials are extruded from separate extruders into the single sheet die which delivers a multi-layer sheet with uniform layer thickness at die exit. This multi-layer sheet is then stretched between counter-rotating rolls to obtain final uniform multi-layer sheet. There are many factors which can influence this process. In this article, calendering a single layer Newtonian or Non-Newtonian material has been extended to analyze a two-layer calendering process for an incompressible Viscoplastic and Newtonian fluids as upper and lower layers with different viscosity ratios. To simplify the equations of motion, the lubrication approximation theory is used. The expressions of non-dimensional pressure gradient, pressure and velocity distribution of both layers are obtained analytically by using proper no slip boundary conditions and dimensionless variables. The dimensionless detachment point is approximated by Regula-Falsi (false position) method. The important engineering factors including detachment point, calendered sheet thickness, roll separation force, power input by rolls, torque on each roll, and adiabatic temperature are all computed. In addition, how the viscosity ratios and viscoplastic casson parameter affect these factors have been investigated. Moreover all established results in literature for single layer calendering Newtonian fluids are also validated at casson parameter β tending towards infinity.

DNS of compressible turbulent flow in convergent-divergent nozzles with isothermal wall

Compressible turbulent flow through canonical convergent-divergent nozzles with circular cross-section, isothermal walls, with and without bell-shaped divergent portion is investigated by means of Direct numerical simulations. The goal of the present study is to demonstrate the effects of flow expansion (and local flow compression) in these nozzles, which have substantial transonic regions, on the mean flow and the turbulence structure. The efficacy of the kinematic or incompressible displacement thickness as a possible lengthscale for computing nondimensional pressure gradient parameters which are indicators of self-similar regions in these flows is shown. Elongation of near-wall flow structures and the dominance of vortex stretching as local flow topology are found in regions of expansion. This is accompanied by a decrease in Reynolds stresses. Regions of local flow compression are found in the divergent portion of the bell-shaped nozzle where vortex compression is the dominant flow topology. A local increase in the streamwise Reynolds stress is found in this region.

Thermal entry flow problem for Giesekus fluid inside an axis-symmetric tube through isothermal wall condition: a comparative numerical study between exact and approximate solution

AbstractThe thermal entry flow problem also known as the Graetz problem is investigated for a Giesekus fluid model. Both analytical (exact) and approximate solutions for velocity are obtained. The nondimensional pressure gradient is numerically obtained via the mean flow rate relation. The energy equation along with the Giesekus fluid velocity is analytically solved for the constant wall temperature case by using the classical separation of variable method. This method transforms the energy equation into a Sturm–Liouville (SL) boundary value problem. The MATLAB solver bvp5c is employed to compute the eigenvalues and the related eigenfunctions numerically. The impact of mobility parameter and Weissenberg number on local Nusselt number, mean temperature, and average Nusselt number is discussed and displayed graphically. It is also found that the presence of the Weissenberg number elevates the Nusselt numbers. Further, the presence of the mobility parameter of the Giesekus fluid model delays the prevalence fully developed conditions in both entrance and fully developed regions. The comparison between approximate and exact solution is also presented. It reveals that both solutions have an exact match with each other for smaller values of mobility parameter and Weissenberg number. However, there is a deviation for larger values of both parameters.

Forced Convection Heat Transfer In a Cylindrical Porous Media exposed to Constant Wall Heat Flux.(Dept.M)

Forced convection heat flow in a cylindrical packed bed is examined numerically. The bed is filled with saturated spherical beads porous media and is exposed to a constant wall heat flux. Besides the energy equation, the generalized from. of the momentum equation including the non-Darcian effects such as the variable porosity, flow inertia, and viscous friction is considered, and the finite difference method is used. The results have been obtained numerically for sphere beads (3≤ d ≤8 mm diameter), ratio of the particle diameter to the pipe radius 0.05 ≤ D ≤0.5, and nondimensional pressure gradient B up to 108. The non-Darcian effects have a significant influence on the behavior of the temperature, thermal entry length. and Nusselt number across and at the channel wall. The channeling phenomenon near the walls enhanced the thermal communication between the fluid/solid matrix. composite and the walls. This fact yielded an overall 21 percent increase in the value of the Nu in the fully developed region, compared to the value predicted when the Darcy model was used. The results gives complete information about the flow structure and heat transfer for the expressed ranges of parameters. Also, useful correlations reporting the dependence of the thermal entry length on the problem parameters (d, D, B) were reported. A direct dependence of the thermal entry length on Re exists and gives the same correlation that obtained for the pure fluid flow case: XE = 0.1 Re To verifY the numerical results a comparison have been done with the numerical results obtained by Poulikakos and Renken [11], Kays and Crawford [26] and Petukhov [27]. The comparison shows a very good agreement for the presented results and proves the validity of the model.

Forced Convective Laminar flow In Elliptic Pipes of Different Aspect Ratios.(Dept.M)

The present analysis investigates non-Darcy forced convective heat transfer in a cylindrical pipe filled with spherical beads saturated with non-Newtonian drag reducing fluid. The cylindrical pipe is subjected to either a uniform heat flux (UHF) or a constant wall temperature (CWT). In modeling the flow, both the energy equation and a modified momentum (Darcy-Forschheimer-Brinkman) equation are used , in which the variable porosity, flow inertia, Brinkman viscous friction (non-Darcian effects) besides the elongational viscosity of drag reducing fluids are taken into consideration and the finite difference technique is used. The results are obtained for a non-Newtonian drag parameter range of 0≤up≤5000, and nondimensional pressure gradient B up to 108. The results show that the non-Newtonian character of drag reducing fluids have a significant influence on the entrance length, beat transfer characteristics and the temperature profiles. Important results documenting and analyzing the behavior of the entrance length and the heat characteristics and its dependence on the non-Newtonian drag parameter are also reported. To examine the adequacy of the present heat transfer model. The governing equations were solved under conditions corresponding to those in Poulikakos and Renken [20]. The comparison of the calculated Nuf with that of Poulikakos and Renken[20]for d=3 and 5mm,up=0.0.05≤D≤0.15 and 104≤B≤106 shows good agreement and validates the presented heat transfer model.

Non-Newtonian Drag Reducing Fluid Flow in a Circular Pipe Filled with Porous Medium in The Non-Darcian Effects.(Dept.M)

This work presents an analysis of a non-Newtonian drag reducing fluid (dilute polymer solutions) in a circular pipe filled with porous media. The flow is developed by modifying the momentum equation for the flow in porous media to account for the elougational viscosity of drag reducing fluids. The modified Darcy-Forschheimer-Brinkman's equation is solved using the finite difference method. The results are obtained for flow Reynolds number up to 105, a non-Newtonian drag parameter range of 0≤ up < /strong>≤5000, and nondimensional pressure gradient B up to 1010. The results show that the non-Newtonian effect of drag reducing fluids have a significant influence on the velocity and signifies relative increase in the maguitude of the velocity in the region adjacent to the wall which in turns signifies the channeling effect. This phenomena is reflected in the great influence on the fluid flow characteristics such as the boundary frictional drag, the elongational viscous drag, and in turns in the total drag. Important results documenting and analyzing the behavior of the velocity and the fluid flow characteristics and its dependence on the non-Newtonian drag parameter are also reported in the course of the study. An experimental investigation was carried out for the flow of water and a dilute polymer solutions with concentrations C = 1, 5, 20, 50 and 100 wppm of the polyacryiamide in a circular rube of 20mm diameter filled with 3.2 mm diameter stainless-steel spheres. Comparison of the numerical results with the experimental results show good agreement of the presented results and prove the validity of the model.

Hydrothermal Behavior of Transient Fluid Flow and Heat Transfer Through a Rotating Curved Rectangular Duct with Natural and Forced Convection

The present work explores a spectral-based computational study on hydrothermal behavior of transient fluid flow with natural and forced convective heat transfer through a rotating curved rectangular duct of strong curvature. The outer wall of the duct is heated while the inner wall cooled, the other walls being thermally insulated. The system rotates about the vertical axis in the positive and negative direction for the Taylor number (-1000≤Tr≤1000) with a constant non-dimensional pressure gradient force, the Dean number Dn = 2000. Time-history analysis is performed and fluid characteristics are well determined by depicting the phase space of the time-history result. It is found that the chaotic flow turns into steady-state flow through multi-periodic and periodic oscillating flows, if Tr is increased either in the positive or in the negative direction. Streamlines of secondary flow and isotherms are obtained at some specific values of Tr, and it is found that the time-dependent flow consists of asymmetric 2- to multi-vortex solutions. Vortex structure of secondary flows is obtained for physically realizable solutions and it is found that maximum 8-vortex is obtained for the chaotic solution while 2-vortex for the steady-state solution. Nusselt number as well as temperature gradient is calculated as an index of heat transfer, and it is found that convective heat transfer is significantly enhanced by the secondary flow; and the chaotic flow, which occurs relatively at small Tr, boosts heat transfer more effectively than the steady-state or other solutions. Finally, our numerical results have been validated with the experimental outcomes and it is found that there is a good agreement between the numerical and experimental investigations.

Hydrodynamic Features of an Undular Bore Traveling on a 1:20 Sloping Beach

The hydrodynamic characteristics, including local and convective accelerations as well as pressure gradient in the horizontal direction, of the external stream of an undular bore propagating on a 1:20 sloping beach are experimentally studied. A bore with the water depth ratio of 1.70 was generated downstream of a suddenly lifted gate. A high-speed particle image velocimetry was employed to measure the velocity fields during the run-up and run-down motions. The time series of free surface elevation and velocity field/profile of the generated bore, comprising a pure bore accompanied by a series of dispersive leading waves, are first demonstrated. Based on the fast Fourier transform (FFT) and inverse FFT (IFFT) techniques, the free surface elevation of leading waves and the counterpart of pure bore are acquired separately at a specified measuring section (SMS), together with the uniform horizontal velocity of the pure bore. The effect of leading-wave-induced velocity shift on the velocity profiles of the generated bore are then evaluated at the SMS. To understand the calculation procedure of accelerations and pressure gradient, three tabulated forms are provided as illustrative examples. Accordingly, the relationships among the partially depth-averaged values of the non-dimensional local acceleration, convective acceleration, total acceleration and pressure gradient of the generated/pure bore acquired at the SMS versus the non-dimensional time are elucidated. The trends in the non-dimensional accelerations and pressure gradient of the external stream of generated bore are compared with those of the pure bore. During the run-up motion from the instant of arrival of the bore front to the moment of the peak level at the SMS, continuous decrease in the onshore uniform horizontal velocity, and successive deceleration of the pure bore in the onshore direction are evidenced, exhibiting the pure bore under the adverse pressure gradient with decreasing magnitude. However, the pure bore once ridden by the leading waves is decelerated/accelerated spatially and accelerated/decelerated temporally in the onshore direction during the rising/descending free surface of each leading wave. This fact highlights the effect of pre-passing/post-passing of the leading wave crest on the velocity distribution of generated bore. It is also found that, although the leading waves have minor contribution on the power spectrum of the free surface elevation as compared with that of the pure bore, the leading waves do play an important role on the magnitudes of both accelerations and pressure gradient. The largest magnitude of the acceleration contributed by the leading waves is around 26 times the counterpart contributed by the pure bore. Further, during the run-down motion right after the moment for the peak level of the bore, a linear increase in the magnitude of the offshore uniform horizontal velocity and a constant local acceleration with increasing time are both identified. The partially depth-averaged value of the non-dimensional pressure gradient is equal to a small negative constant (−0.0115) in the offshore direction, indicating that the bore is subject to a constant favorable pressure gradient.

Characteristics of Accelerations and Pressure Gradient during Run-Down of Solitary Wave over Very Steep Beach: A Case Study

An experimental investigation is performed to elucidate the variations of accelerations and pressure gradients in the external stream of retreating flow during the run-down phase of a non-breaking solitary wave, propagating over a 1:3 sloping beach. Two solitary waves that have the incident wave heights (H0) of 2.9 and 5.8 cm, with respective still water depths (h0) of 8.0 and 16.0 cm (Cases A and B), were generated in a wave flume, resulting in the incident wave-height to water-depth ratios (H0/h0) being identically equal to 0.363. The latter case was only used to highlight the non-dimensional features of the wave celerity, the time history of horizontal velocity and the breaker type, which all exhibit similarity to those of the former. Two flow visualization techniques such as particle trajectory method and fluorescent dye strip and a high-speed particle image velocimetry (HSPIV) were utilized to provide the flow images and velocity fields. Based on the ensemble-averaged velocity fields and profiles, the partially depth-averaged (i.e., excluding the part in the boundary layer) values of accelerations and pressure gradient at a specified measuring section are then smoothed by a symmetric five-point smoothing scheme. Eventually, the smoothed values of the accelerations and pressure gradient are used to highlight the dynamic features of the external stream of retreating flow. It is found that, at the section of the incipient flow separation, the non-dimensional local acceleration (with respect to the gravity acceleration) in the offshore direction keeps increasing from the moment at which the run-up motion ends to the counterpart at which the incipient flow separation occurs. Afterwards, growth of the primary vortex develops with its core translating offshore. The corresponding non-dimensional local acceleration at the (moving) core section increases to a maximum of around 1.0 at the instant for occurrence of the hydraulic jump with abrupt rise of the free surface; and then decreases to zero at time for transformation of the curling jet into the projecting jet. The results exhibit that the external stream of retreating flow is accelerated temporally in the offshore direction for the interval between the time for the end of run-up motion and that for the formation of projecting jet. However, for later time interval up to generation of the two-phase flow field, the non-dimensional local acceleration in the offshore direction varies from zero to a negative maximum of −2.117 at the moment for the projecting jet heading downward before the impingement. It then decreases in magnitude continuously. The trend reveals that the external stream is decelerated temporally in the offshore direction for this later time interval. Further, at the section of the incipient flow separation, the non-dimensional pressure gradient (also with reference to the gravity acceleration) in the offshore direction increases from 0.225 for the time at which the run-up motion ends to 0.721 for the instant at which the incipient flow separation takes place. The trend highlights the external stream being under increasing adverse pressure gradient and more decelerated spatially with the increasing time, thus resulting in occurrence of the incipient flow separation. Afterwards, the value of the non-dimensional pressure gradient keeps increasing and eventually reaches a positive maximum of 2.011 and then decreases consecutively until the two-phase flow field is generated. In addition, due to the influence of acceleration of the external stream in the offshore direction, the non-dimensional vorticity of primary vortex core increases with increasing time up to the moment for occurrence of the projecting jet. Nevertheless, the non-dimensional vorticity of primary vortex core keeps decreasing with increasing time T for the later time interval due to the influence of deceleration of the external stream in the offshore direction. Finally, considerably large magnitudes of the non-dimensional accelerations and pressure gradient greater than 1.5 taking place at two non-dimensional times are worthy of noting. The negative maximum value of the non-dimensional convective acceleration equal to −2.005 appears at the instant for the occurrence of hydraulic jump. In addition, the negative maximum values of the non-dimensional local acceleration, total acceleration and pressure gradient unexpectedly as high as −2.117, −1.694 and 2.011, respectively, appear simultaneously at time for the projecting jet heading towards the retreating free surface. Under such a situation, the external stream of retreating flow is highly decelerated in the offshore direction under the fairly large adverse pressure gradient, thus forcing the retreating flow to move upwards rapidly. Meanwhile, the non-dimensional local acceleration in the vertical direction is surprisingly found to be 3.37. The result strongly reconfirms the evident rise of the free surface in the vicinity of the core section and reveals very rapid change from negative, via nearly zero, to positive vertical velocity.

Couette-Poiseuille flow based non-linear flow over a square cylinder near plane wall

A numerical study on the flow over a square cylinder in the vicinity of a wall is conducted for different Couette-Poiseuille-based non-uniform flow with the non-dimensional pressure gradient P varying from 0 to 5. The non-dimensional gap ratio L (=H/a) is changed from 0.1 to 2, where H is gap height between the cylinder and wall, and a is the cylinder width. The governing equations are solved numerically through finite volume method based on SIMPLE algorithm on a staggered grid system. Both P and L have a substantial influence on the flow structure, time-mean drag coefficient , fluctuating (rms) lift coefficient (CL), and Strouhal number St. The changes in P and L leads to four distinct flow regimes (I, II, III and IV). Following the flow structure change, the , CL, and St all vary greatly with the change in L and/or P. The and CL both grow with increasing P and/or L. The St increases with P for a given L, being less sensitive to L for a smaller P ( 2). A strong relationship is observed between the flow regimes and the values of , CL and St. An increase in P affects the pressure distribution more on the top surface than on bottom surface while an increase in L does the opposite.

Reynolds stress structures in a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation.

Mean Reynolds stress profiles and instantaneous Reynolds stress structures are investigated in a self-similar adverse pressure gradient turbulent boundary layer (APG-TBL) at the verge of separation using data from direct numerical simulations. The use of a self-similar APG-TBL provides a flow domain in which the flow gradually approaches a constant non-dimensional pressure gradient, resulting in a flow in which the relative contribution of each term in the governing equations is independent of streamwise position over a domain larger than two boundary layer thickness. This allows the flow structures to undergo a development that is less dependent on the upstream flow history when compared to more rapidly decelerated boundary layers. This APG-TBL maintains an almost constant shape factor of H = 2.3 to 2.35 over a momentum thickness based Reynolds number range of Reδ2 = 8420 to 12400. In the APG-TBL the production of turbulent kinetic energy is still mostly due to the correlation of streamwise and wall-normal fluctuations, 〈uv〉, however the contribution form the other components of the Reynolds stress tensor are no longer negligible. Statistical properties associated with the scale and location of sweeps and ejections in this APG-TBL are compared with those of a zero pressure gradient turbulent boundary layer developing from the same inlet profile, resulting in momentum thickness based range of Reδ2 = 3400 to 3770. In the APG-TBL the peak in both the mean Reynolds stress and the production of turbulent kinetic energy move from the near wall region out to a point consistent with the displacement thickness height. This is associated with a narrower distribution of the Reynolds stress and a 1.6 times higher relative number of wall-detached negative uv structures. These structures occupy 5 times less of the boundary layer volume and show a similar reduction in their streamwise extent with respect to the boundary layer thickness. A significantly lower percentage of wall-attached structures is observed in the present case when compared with a similar investigation of a rapidly decelerating APG-TBL, suggesting that these wall-attached features could be the remanent from the lower pressure gradient domain upstream.

Large-eddy simulation of flow over a grooved cylinder up to transcritical Reynolds numbers

We report wall-resolved large-eddy simulation (LES) of flow over a grooved cylinder up to the transcritical regime. The stretched-vortex subgrid-scale model is embedded in a general fourth-order finite-difference code discretization on a curvilinear mesh. In the present study $32$ grooves are equally distributed around the circumference of the cylinder, each of sinusoidal shape with height $\unicode[STIX]{x1D716}$, invariant in the spanwise direction. Based on the two parameters, $\unicode[STIX]{x1D716}/D$ and the Reynolds number $Re_{D}=U_{\infty }D/\unicode[STIX]{x1D708}$ where $U_{\infty }$ is the free-stream velocity, $D$ the diameter of the cylinder and $\unicode[STIX]{x1D708}$ the kinematic viscosity, two main sets of simulations are described. The first set varies $\unicode[STIX]{x1D716}/D$ from $0$ to $1/32$ while fixing $Re_{D}=3.9\times 10^{3}$. We study the flow deviation from the smooth-cylinder case, with emphasis on several important statistics such as the length of the mean-flow recirculation bubble $L_{B}$, the pressure coefficient $C_{p}$, the skin-friction coefficient $C_{f\unicode[STIX]{x1D703}}$ and the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$. It is found that, with increasing $\unicode[STIX]{x1D716}/D$ at fixed $Re_{D}$, some properties of the mean flow behave somewhat similarly to changes in the smooth-cylinder flow when $Re_{D}$ is increased. This includes shrinking $L_{B}$ and nearly constant minimum pressure coefficient. In contrast, while the non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}$ remains nearly constant for the front part of the smooth cylinder flow, $\unicode[STIX]{x1D6FD}$ shows an oscillatory variation for the grooved-cylinder case. The second main set of LES varies $Re_{D}$ from $3.9\times 10^{3}$ to $6\times 10^{4}$ with fixed $\unicode[STIX]{x1D716}/D=1/32$. It is found that this $Re_{D}$ range spans the subcritical and supercritical regimes and reaches the beginning of the transcritical flow regime. Mean-flow properties are diagnosed and compared with available experimental data including $C_{p}$ and the drag coefficient $C_{D}$. The timewise variation of the lift and drag coefficients are also studied to elucidate the transition among three regimes. Instantaneous images of the surface, skin-friction vector field and also of the three-dimensional Q-criterion field are utilized to further understand the dynamics of the near-surface flow structures and vortex shedding. Comparison of the grooved-cylinder flow with the equivalent flow over a smooth-wall cylinder shows structural similarities but significant differences. Both flows exhibit a clear common signature, which is the formation of mean-flow secondary separation bubbles that transform to other local flow features upstream of the main separation region (prior separation bubbles) as $Re_{D}$ is increased through the respective drag crises. Based on these similarities it is hypothesized that the drag crises known to occur for flow past a cylinder with different surface topographies is the result of a change in the global flow state generated by an interaction of primary flow separation with secondary flow recirculating motions that manifest as a mean-flow secondary bubble. For the smooth-wall flow this is accompanied by local boundary-layer flow transition to turbulence and a strong drag crisis, while for the grooved-cylinder case the flow remains laminar but unsteady through its drag crisis and into the early transcritical flow range.

Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer at the verge of separation

The statistical properties are presented for the direct numerical simulation of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) at the verge of separation. The APG TBL has a momentum thickness-based Reynolds number range from $Re_{\unicode[STIX]{x1D6FF}_{2}}=570$ to 13 800, with a self-similar region from $Re_{\unicode[STIX]{x1D6FF}_{2}}=10\,000$ to 12 300. Within this domain the average non-dimensional pressure gradient parameter $\unicode[STIX]{x1D6FD}=39$, where for a unit density $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FF}_{1}P_{\!e}^{\prime }/\unicode[STIX]{x1D70F}_{w}$, with $\unicode[STIX]{x1D6FF}_{1}$ the displacement thickness, $\unicode[STIX]{x1D70F}_{w}$ the mean shear stress at the wall and $P_{\!e}^{\prime }$ the far-field pressure gradient. This flow is compared with previous zero pressure gradient and mild APG TBL ($\unicode[STIX]{x1D6FD}=1$) results of similar Reynolds number. All flows are generated via the direct numerical simulation of a TBL on a flat surface with far-field boundary conditions tailored to apply the desired pressure gradient. The conditions for self-similarity, and the appropriate length and velocity scales, are derived. The mean and Reynolds stress profiles are shown to collapse when non-dimensionalised on the basis of these length and velocity scales. As the pressure gradient increases, the extent of the wake region in the mean streamwise velocity profiles increases, whilst the extent of the log-layer and viscous sublayer decreases. The Reynolds stress, production and dissipation profiles of the APG TBL cases exhibit a second outer peak, which becomes more pronounced and more spatially localised with increasing pressure gradient. This outer peak is located at the point of inflection of the mean velocity profiles, and is suggestive of the presence of a shear flow instability. The maximum streamwise velocity variance is located at a wall normal position of $\unicode[STIX]{x1D6FF}_{1}$ of spanwise wavelength of $2\unicode[STIX]{x1D6FF}_{1}$. In summary as the pressure gradient increases the flow has properties less like a zero pressure gradient TBL and more akin to a free shear layer.

The minimal-span channel for rough-wall turbulent flows

Roughness predominantly alters the near-wall region of turbulent flow while the outer layer remains similar with respect to the wall shear stress. This makes it a prime candidate for the minimal-span channel, which only captures the near-wall flow by restricting the spanwise channel width to be of the order of a few hundred viscous units. Recently, Chung et al. (J. Fluid Mech., vol. 773, 2015, pp. 418–431) showed that a minimal-span channel can accurately characterise the hydraulic behaviour of roughness. Following this, we aim to investigate the fundamental dynamics of the minimal-span channel framework with an eye towards further improving performance. The streamwise domain length of the channel is investigated with the minimum length found to be three times the spanwise width or 1000 viscous units, whichever is longer. The outer layer of the minimal channel is inherently unphysical and as such alterations to it can be performed so long as the near-wall flow, which is the same as in a full-span channel, remains unchanged. Firstly, a half-height (open) channel with slip wall is shown to reproduce the near-wall behaviour seen in a standard channel, but with half the number of grid points. Next, a forcing model is introduced into the outer layer of a half-height channel. This reduces the high streamwise velocity associated with the minimal channel and allows for a larger computational time step. Finally, an investigation is conducted to see if varying the roughness Reynolds number with time is a feasible method for obtaining the full hydraulic behaviour of a rough surface. Currently, multiple steady simulations at fixed roughness Reynolds numbers are needed to obtain this behaviour. The results indicate that the non-dimensional pressure gradient parameter must be kept below 0.03–0.07 to ensure that pressure gradient effects do not lead to an inaccurate roughness function. An empirical costing argument is developed to determine the cost in terms of CPU hours of minimal-span channel simulations a priori. This argument involves counting the number of eddy lifespans in the channel, which is then related to the statistical uncertainty of the streamwise velocity. For a given statistical uncertainty in the roughness function, this can then be used to determine the simulation run time. Following this, a finite-volume code with a body-fitted grid is used to determine the roughness function for square-based pyramids using the above insights. Comparisons to experimental studies for the same roughness geometry are made and good agreement is observed.

Effects of horizontal pressure gradients on bed destabilization under waves

We report on new experiments designed to investigate bed destabilization processes in a two-dimensional wave flume physical model of a beach. The mobile bed consists of non-cohesive granular material of low density. The wave conditions are provided by repeating a cycle of waves made of two bichromatic groups of different period. The horizontal and vertical velocities are acoustically profiled vertically from free-stream elevation down to the still bed level in the surf zone. Additional measurements of the fluid pressure at positions closely aligned horizontally and vertically in and slightly above the sediment bed are undertaken. Mobile bed interfaces, still bed and top interface, are detected via acoustic and optical methods. Both methods are cross-compared and give similar results. Flow turbulence over the bed is analysed, the Reynolds turbulent shear stress is found negligible compared to the orbital flow induced momentum diffusion. The shear stress and the horizontal pressure gradient are computed at near-bed elevation and used in the bed incipient plug flow model of Sleath (Cont. Shelf Res., vol. 19 (13), 1999, pp. 1643–1664). Both the model and the measurements confirm that destabilization occurs when the non-dimensional pressure gradient (or Sleath number) exceeds the threshold value of 0.3 which is simultaneous with strong flow acceleration. The near-bottom fluid shear stress detected during these flow accelerations at steep wave fronts is found experimentally to be negative, which is retrieved with an unsteady plug flow model. This is suggesting that the fluid above the bed resists the sediment layer motion at these particular phases.

Nondimensional maximum pressure gradient of tunnel compression waves generated by offset running axisymmetric trains

A high-speed train entering a tunnel generates a compression wave in the tunnel and a micro-pressure wave. The magnitude of the micro-pressure wave is approximately proportional to the maximum pressure gradient of the compression wave (∂p/∂t)max. High-speed railway operations require the consideration of the nose shape of the train in order to reduce (∂p/∂t)max generated by a train entering a tunnel. In this study, the compression waves generated by axisymmetric trains running at the offset position in double-track tunnels were investigated using a train launcher facility. Paraboloids of revolution, ellipsoids of revolution, and cones were used as the simplest nose shapes. The cone-nose train generated the largest value of (∂p/∂t)max, and the paraboloid-nose train generated the smallest value of (∂p/∂t)max among the three nose shapes. Although this tendency is the same as that for center running, the ratio of (∂p/∂t)max for the cone-nose train to that for the paraboloid-nose train became larger for offset running than center running. Moreover, the maximum value for the generation time of the compression wave was derived from acoustic theory, and the nondimensionalization of (∂p/∂t)max considering the nose length was proposed using it. Since the inverse of the nondimensionalized (∂p/∂t)max denotes the ratio of the generation time of the compression wave to its maximum value, it was defined as the efficiency of a train nose in the range of zero to unity. The values of the efficiency were almost constant with the nose length and 0.7, 0.6, and 0.5 for paraboloid-, ellipsoid-, and cone-nose shapes, respectively.

Direct numerical simulation of a self-similar adverse pressure gradient turbulent boundary layer

The statistical properties of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) are presented. The flow is generated via the direct numerical simulation of a TBL on a flat surface with a farfield boundary condition designed to apply the desired pressure gradient. The conditions for self-similarity and appropriate scaling are derived, with the mean profiles, Reynolds stress profiles, and turbulent kinetic energy budgets non-dimensionalised using this scaling. The APG TBL has a momentum thickness based Reynolds number range from Reδ2=300 to 6000, with a self-similar region spanning a Reynolds number range from Reδ2=3500 to 4800. Within this range the non-dimensional pressure gradient parameter β=1. Two-point correlations of each of the velocity components in the streamwise/wall-normal plane are also presented, which illustrate the statistical imprint of the structures in this plane for the APG TBL.

Electroosmotic flow through a microtube with sinusoidal roughness

In this paper, a perturbation method is introduced to study the electroosmotic flow (EOF) in a microtube with slightly corrugated wall. The corrugation of the wall is described as periodic sinusoidal wave with small amplitude. Based on linearized Poisson–Boltzmann equation and the Cauchy momentum equation, the perturbation solutions for velocity and volume flow rate are obtained. The influences of the amplitude δ, the wave number λ, the nondimensional electrokinetic width K and the nondimensional pressure gradient G on the velocity and flow rate are analyzed graphically and discussed in detail. The results show that the flow rate Q of the EOF through a corrugated channel tends to the flow rate Q0 of the EOF through a smooth channel when amplitude δ tends to zero, but the flow rate Q is always smaller than the flow rate Q0 in the smooth channel. The flow retardation of the roughness on the flow rate of the EOF always increases with the augment of the nondimensional electrokinetic width K.

Interactions of Vortices of a Square Cylinder and a Rectangular Vortex Generator Under Couette–Poiseuille Flow

This study focuses on interactions of vortices generated by a family of eddy-promoting upstream rectangular cylinders (of different heights a* and widths b*) with the shear layers of a downstream square cylinder (of height A*) placed near a plane in an in-line tandem arrangement under the incidence of Couette–Poiseuille flow based nonuniform linear/nonlinear velocity profile. The dimensionless operational parameters are cylinders spacing distance S, ratio of heights r2=a*/A* (≤1), aspect ratio r1=b*/a* (≤1), Reynolds number Re (based on the velocity at height A* for Couette flow), ReU2 (based on the velocity at height 10A* for Couette–Poiseuille flow), and nondimensional pressure gradient P at the inlet. The governing equations are solved numerically through a pressure-correction-based iterative algorithm (SIMPLE) with the quadratic upwind interpolation for convective kinematics (QUICK) scheme for convective terms. The major issue of appearing multiple peaks in the spectrum of the fluctuating lift coefficient of the downstream cylinder is addressed and justified exhibiting the flow patterns. While considering the rectangular shape (for the upstream cylinder) and nonlinear velocity (at the inlet), the possibility of generating the unsteadiness in the steady wake flow of the downstream cylinder at a Re (based on height a*) less than the critical Re for the downstream cylinder is documented here. The dependence of flow characteristics of the downstream cylinder on the angle of incident linear velocity at specific S and r1 is also demonstrated here. It is observed that the discontinuous jump in the aerodynamic characteristics (due to a sudden change from one distinct flow pattern to the other in the critical spacing distance regime) is directly proportional to the height of the vortex generator. Increasing P under the same characteristic velocity causes the steady flow of cylinder(s) to convert to a periodic flow and reduces the critical spacing distance for the vortex generator.