Non-dimensional Quantities Research Articles

On stagnation pressure increases in calorically perfect, ideal gases

When stagnation pressure rises in a natural or numerically simulated flow it is frequently a cause for concern, as one usually expects viscosity and turbulence to cause stagnation pressure to decrease. In fact, if stagnation pressure increases, one may suspect measurement or numerical errors. However, this need not be the case, as the laws of nature do not require that stagnation pressure continually decreases. In order to help clarify matters, the objective of this work is to understand the conditions under which stagnation pressure will rise in the unsteady/steady flows of compressible, viscous, calorically perfect, ideal gases. Furthermore, at a more practical level, the goal is to understand the conditions under which stagnation pressure will increase in flows simulated with the Reynolds averaged Navier–Stokes equations and eddy-viscosity turbulence models. In order to provide an improved understanding of increases in stagnation pressure for both these scenarios, transport equations are derived that govern its behavior in the unaveraged and Reynolds averaged settings. These equations are utilized to precisely determine the relationship between changes in stagnation pressure and zeroth, first, and second derivatives of fundamental flow quantities. Furthermore, these equations are utilized to demonstrate the relationship between changes in stagnation pressure and fundamental non-dimensional quantities that govern the conductivity, viscosity, and compressibility of the flow. In addition, based on an analysis of the Reynolds averaged equation (for eddy-viscosity turbulence models), it is shown that stagnation pressure is particularly likely to experience a spurious rise at the outer edges of shear layers that are undergoing convex curvature. Thereafter, numerical experiments are performed which confirm the primary aspects of the theoretical analysis.

Transverse vibration of pipe conveying fluid made of functionally graded materials using a symplectic method

Problems related to the transverse vibration of pipe conveying fluid made of functionally graded material (FGM) are addressed. Based on inside and outside surface material compositions of the pipe, FGM pipe conveying fluid can be classified into two cases. It is hypothesized that the physical parameters of the material along the direction of the pipe wall thickness change in the simple power law. A differential equation of motion expressed in non-dimensional quantities is derived by using Hamilton's principle for systems of changing mass. Using the assuming modal method, the pipe deflection function is expanded into a series, in which each term is expressed to admissible function multiplied by generalized coordinate. Then, the differential equation of motion is discretized into the two order differential equations expressed in the generalized coordinates. Based on symplectic elastic theory and the introduction of dual system and dual variable, Hamilton's dual equations are derived, and the original problem is reduced to eigenvalue and eigenvector problem in the symplectic space. Finally, a symplectic method is employed to analyze the vibration and stability of FGM pipe conveying fluid. For a clamped–clamped FGM pipe conveying fluid in “case 1” and “case 2”, the dimensionless critical flow velocity for first-mode divergence and the critical coupled-mode flutter flow velocity are obtained, and the variations between the real part and imaginary part of dimensionless complex frequency and fluid velocity, mass ratio and the power law exponent (or graded index, volume fraction) for FGM pipe conveying fluid are analyzed.

Modeling and analysis of free vibrations in thermoelastic hollow spheres

Purpose– The purpose of this paper is to present a model to analyze free vibrations in a transradially isotropic, thermoelastic hollow sphere subjected to stress free, thermally insulated or stress free, isothermal and rigidly fixed, thermally insulated or rigidly fixed, isothermal boundary conditions.Design/methodology/approach– The potential functions along with spherical wave solution have been used to reduce the system of governing partial differential equations to a coupled system of ordinary differential equations in radial coordinates after employing non-dimensional quantities. Matrix Frobenius method of extended power series has been employed to obtain accurate solution of coupled differential equations in terms of radial coordinates. The mathematical model of the considered problem has been solved analytically to obtain the characteristics equations after imposing the appropriate boundary conditions at the outer and inner surfaces of the hollow sphere. The characteristic equations which govern various types of vibration modes expected to exist have been derived in the compact form. The special cases of spheroidal and toroidal modes of vibrations have been deduced from the characteristic equations and discussed.Findings– The toroidal mode has been found to be independent of temperature change. The magnitude of lowest frequency and damping factor are significantly affected in the presence of thermal field and increase with an increase in the spherical harmonics in addition to geometry of the structure.Originality/value– The matrix Frobenius method has been used to develop analytical solutions and functional iteration technique to carry out numerical simulations of such structures for the first time. The simulated results are presented graphically and compared with the available literature.

Turbulent drag reduction by spanwise traveling ribbed surface waves

The combination of passive and active drag reduction techniques is investigated by particle-image velocimetry (PIV) and micro-particle tracking velocimetry (μ-PTV) measurements. To be more precise, the impact of a riblet-structured surface, which represents the passive control technique, undergoing spanwise transversal wave motion, which defines the active flow control method, on the wall-shear stress distribution of a zero-pressure gradient turbulent boundary layer is analyzed. The experimental setup consists of a flat plate equipped with an insert to generate a spanwise traveling wave on a riblet-structured aluminum sheet. The PIV and μ-PTV measurements are conducted downstream of the actuated riblet surface at two momentum thickness based Reynolds numbers Reθ=1200 and 2080. The surface deformation is generated by an electromagnetic actuator system at three amplitudes A= 0.25, 0.30, and 0.375 mm with a wavelength of λ=160 mm and a frequency of f=81 Hz. The nondimensional quantities in inner wall units defined by the friction velocity of the non-actuated flat plate flow at Reθ=1200 are the amplitudes A+=6, 7, and 9, the wavelength λ+=3862, and the frequency T+=110 and the corresponding quantities for Reθ=2800 are A+=11, 14 and 17, λ+=7170, and T+=380. The results are twofold. On the one hand, the combination of the riblet surface with the transversal wave motion increases the local drag reduction (DR). That is, compared to a boundary layer flow over a non-actuated smooth surface a local drag reduction of up to DR =9.4 % is determined at the chosen measurement position whereas the drag reduction due to the riblet surface is DR =4.7%. On the other hand, the dependence of the local drag reducing efficiency of the riblets on their geometric dimensions with respect to the flow field could be decreased. That is, the transversal wave motion extends the positive drag reducing efficiency range of the riblets.

Response of a fluid-immersed microcantilever close to a deformable body

The importance of hydrodynamics upon the response of a microcantilever immersed in a viscous fluid has been well established [J. E. Sader, J. Appl. Phys. 84, 64 (1998); C. A. Eysden and J. E. Sader, J. Appl. Phys. 101, 044908 (2007)]. It has previously been shown that the presence of a nearby rigid planar surface can significantly alter a microcantilever's non-contact response, through microcantilever–surface hydrodynamic interactions [C. P. Green and J. E. Sader, Phys. Fluids 17, 073102 (2005); C. P. Green and J. E. Sader, J. Appl. Phys. 98, 114913 (2005); R. J. Clarke et al., J. Fluid Mech. 545, 397426 (2005); R. J. Clarke et al., Phys. Rev. Lett. 96, 050801 (2006).]. In cases where the nearby surface is a finite-sized deformable body, such as in noncontact microrheology measurements, we expect to see further changes in the microcantilever's response. Hence, we here compute the thermal spectra of several microcantilevers in the presence of different compliant samples that have the characteristics of soft biological fibres. Our findings demonstrate that the elastohydrodynamic regime can substantially dictate the extent to which the compliance of a given body is evident in the microcantilever's thermal spectra, and suggest that certain nondimensional quantities should lie within particular, ranges for this to be the case. We expect these findings to be of interest in areas such as Atomic Force Microscopy, microsensing, and non-contact microrheology.

Modelling thermal recycling occurring in groundwater heat pumps (GWHPs)

The performance of a Ground Water Heat Pump (GWHP) is often impaired by the thermal recycling between the injection and the extraction well(s), and hence this phenomenon should be evaluated in the design of open loop geothermal plants. The numerical flow and heat transport simulation of a GWHP requires an expensive characterization of the aquifer to obtain reliable input data, which is usually not affordable for small installations. To provide a simple, fast and inexpensive tool for preliminary and sensitivity analyses, an open-source numerical code was developed, which solves the hydraulic and thermal transport problem of a well doublet in the presence of a subsurface flow. The code, called TRS (Thermal Recycling Simulator), is based on a finite-difference approximation of the potential flow theory. The method was validated through the comparison with flow and heat transport simulations with FEFLOW. Subsequently, TRS was run with different values of the aquifer and plant parameters. The correlation observed between some characteristic non-dimensional quantities permitted an empirical correlation to be developed, that describes the time evolution of the extracted water temperature. An example is given for the use of the numerical code and the formula in the dimensioning of an open loop geothermal plant.

A Computational Analysis of the Effect of Mass and Radiative Heat Transfer on Free Convective Boundary Layer Flow over a Vertical Plate

The effects of mass and radiative heat transfer on free convective flow of a viscous incompressible optically thick fluid towards a vertical surface have been investigated. The nonlinear non-dimensional, similarity-transformed boundary-layer equations governing the problem are solved using an efficient numerical method based on the Runge-Kutta integration scheme and shooting iteration technique. Numerical calculations were carried out for different values of the various non-dimensional quantities governing the flow regime. The analysis shows that the temperature decreases with increasing radiation parameter, N while an increase in the Prandtl number leads to a corresponding decrease in the temperature profile; a rise in the thermal Grasshof and the mass transfer number leads to increase in the velocity profile and a rise in the Schmidt number Sc leads to a decrease in the concentration profile.

A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape

[2] Accurate estimates of mass-exchange parameters in transient storage zones are needed to better understand and quantify solute transport and dispersion in riverine systems. Currently, the predictive mean residence time relies on an empirical entrainment coefficient with a range in variance due to the absence of hydraulic and geomorphic quantities driving mass exchange. Two empirically derived relationships are presented for the mean residence time of lateral cavities—a prevalent and widely recognized type of transient storage—in gravel-bed rivers and streams that incorporates hydraulic and geomorphic parameters. The relationships are applicable for gravel-bed rivers and streams with a range of cavity width to length (W/L) aspect ratios (0.2–0.75), shape, and Reynolds numbers (Re, ranging from 1.0 × 104 to 1.0 × 107). The relationships equate normalized mean residence time to nondimensional quantities: Froude number, Re, W/L, depth ratio (ratio of cavity to shear layer depth), roughness factor (ratio of shear to channel velocity), and shape factor (representing degree of cavity equidimensionality). One relationship excludes bed roughness (equation (13)) and the other includes bed roughness (equation (14)). The empirically derived relationships have been verified for conservative tracers (R2 of 0.83) within a range of flow and geometry conditions. Topics warranting future research are testing the empirical relationship that includes the roughness factor using parameters measured in the vicinity of the cavity to reduce the variance in the correlation, and further development of the relationship for nonconservative transport.

Study of dimensionless quantities to analyse front and rear wall of keyhole formed during laser beam welding

Fluid flow mechanisms present in Keyhole (KH) during Laser Beam Welding (LBW) process influence the associated heat and mass transfer. In an attempt to describe these complexities for eventual optimization of LBW parameters, a dimensionless analysis using Mach (Ma), Raleigh (Ra), Reynolds (Re) and Marangoni (Mg) numbers have been carried out. This analysis describes hydrodynamics of melt and vapour phase appearing in the front and rear wall of KH. The non-dimensional hydrodynamic quantities describe the mechanism behind flow pattern present in melt-vapour in terms of ratio of convection–conduction heat transfer occurring within KH. The analysis shows that the higher Marangoni number indicates stronger Marangoni convection in the KH causing relatively higher capillary flow in the melt pool. The laminar-turbulent flow of melt-vapour in KH medium is described in terms of ratio of Reynolds and Mach numbers (Re/Ma). The pressure distribution in the KH accounts for the melt-vapour ejection rate. A relationship between depth and radius of KH has been obtained as a function of delivered laser power.

Heat and mass transfer on a MHD third grade fluid with partial slip flow past an infinite vertical insulated porous plate in a porous medium

The influence of third grade, partial slip and other thermophysical parameters on the steady flow, heat and mass transfer of viscoelastic third grade fluid past an infinite vertical insulated plate subject to suction across the boundary layer has been investigated. The space occupying the fluid is porous. The momentum equation is characterized by a highly nonlinear boundary value problem in which the order of the differential equation exceeds the number of available boundary conditions. An efficient numerical scheme of midpoint technique with Richardson’s extrapolation is employed to solve the governing system of coupled nonlinear equations of momentum, energy and concentration. Numerical calculations were carried out for different values of various interesting non-dimensional quantities in the slip flow regime with heat and mass transfer and were shown with the aid of figures. The values of the wall shear stress, the local rate of heat and mass transfers were obtained and tabulated. The analysis shows that as the fluid becomes more shear thickening, the momentum boundary layer decreases but the thermal boundary layer increases; the magnetic field strength is found to decrease with an increasing temperature distribution when the porous plate is insulated. The consequences of increasing the permeability parameter and Schmidt number decrease both the momentum and concentration boundary layer thicknesses respectively whereas an increase in the thermal Grashof number gives rise to the thermal boundary layer thickness.

Geometry and interaction of structures in homogeneous isotropic turbulence

AbstractA strategy to extract turbulence structures from direct numerical simulation (DNS) data is described along with a systematic analysis of geometry and spatial distribution of the educed structures. A DNS dataset of decaying homogeneous isotropic turbulence at Reynolds number ${\mathit{Re}}_{\lambda } = 141$ is considered. A bandpass filtering procedure is shown to be effective in extracting enstrophy and dissipation structures with their smallest scales matching the filter width, $L$. The geometry of these educed structures is characterized and classified through the use of two non-dimensional quantities, ‘planarity’ and ‘filamentarity’, obtained using the Minkowski functionals. The planarity increases gradually by a small amount as $L$ is decreased, and its narrow variation suggests a nearly circular cross-section for the educed structures. The filamentarity increases significantly as $L$ decreases demonstrating that the educed structures become progressively more tubular. An analysis of the preferential alignment between the filtered strain and vorticity fields reveals that vortical structures of a given scale $L$ are most likely to align with the largest extensional strain at a scale 3–5 times larger than $L$. This is consistent with the classical energy cascade picture, in which vortices of a given scale are stretched by and absorb energy from structures of a somewhat larger scale. The spatial distribution of the educed structures shows that the enstrophy structures at the $5\eta $ scale (where $\eta $ is the Kolmogorov scale) are more concentrated near the ones that are 3–5 times larger, which gives further support to the classical picture. Finally, it is shown by analysing the volume fraction of the educed enstrophy structures that there is a tendency for them to cluster around a larger structure or clusters of larger structures.

A novel perspective to high-speed cross-hot-wire calibration methodology

A practical cross-hot-wire calibration and data reduction methodology for instantaneous measurements of mass flux and flow angle is developed for two dimensional subsonic compressible flows. Historically, data reduction for flow conditions of 0.4 < M < 1.2 is regarded as problematic, even in the simplified case of flow normal mounted wires. Thus, in comparison with the incompressible and supersonic conditions, the literature addressing these flow regimes is quite limited. The present study addresses this void by relating the wire voltages to flow conditions through renormalized, Mach and overheating independent, nondimensional quantities. Therefore, a short and robust calibration can be performed in an unheated free jet facility with applicability toward a broad range of planar flow conditions. This disposes the need for typical closed loop calibration wind tunnels which vary flow velocity, density and temperature independently to parameterize the voltage dependency in a purely empirical manner.

Development of Cryogenic Oscillating Heat Pipe as a New Device for Indirect/Conduction Cooled Superconducting Magnets

Cryogenic oscillating heat pipes (OHPs) have been proposed as a new heat transfer device for conduction/indirect cooling of high-temperature superconducting (HTS) magnets. OHP is a highly effective two-phase heat transfer device which can transport several orders of magnitude greater heat flux than the heat conduction of solids. The performance of cryogenic OHPs has been intensively examined and the results indicate the ability of dramatically improving the performance of HTS magnets. Semi-empirical correlations stating thermo-physical properties of cryogenic OHPs are introduced based on those of room temperature OHPs. The modeling with non-dimensional quantities is useful for the design of cryogenic OHPs.

Finite-difference solution of unsteady natural convection flow past a nonisothermal vertical cone under the influence of a magnetic field and thermal radiation

An analysis is performed to study the heat transfer characteristics of natural convection over a vertical cone under the combined effects of a magnetic field and thermal radiation. The cone surface is subjected to a variable surface temperature. The fluid considered is a gray absorbing/emitting, but non-scattering medium. The boundary layer equations governing the flow are reduced to non-dimensional equations using non-dimensional quantities valid in the free-convection regime. The resulting non-dimensional governing equations are solved by an implicit finite-difference method of the Crank-Nicolson type, which is rapidly convergent and unconditionally stable. Numerical results are obtained for velocity, temperature, local and average skin friction, and local and average Nusselt numbers for various values of parameters occurring in the problem and are presented in the graphical form. Excellent agreement of the results obtained with available data is demonstrated.

Numerical Prediction of Product Deformation and Identification of Major Controlling Factors during the IML Process Including Thermal Impact

The in-mold labeling (IML) process has become a popular choice for the exterior decoration of many mobile devices due to its capability of rich color representation with a relatively easy manufacturing procedure. A molten polymer is injected into a cavity where pre-decorated film has been initially inserted. A heterogeneous characteristic of involved material coupled with various processing parameters can induce defects of the final product such as film delamination, wash-out, and flow-mark. In this study, major controlling parameters for the deformation of the final product have been identified with the design of experiment (DOE) method using several nondimensional quantities of temperature, material property, and thickness ratio. MOLDFLOW and ABAQUS software has been simultaneously used to simulate the injection process and thermal impact process, respectively. We found that the thickness ratio is a critical factor on final deformation, and the material property ratio has a relatively minor effect.

Constitutive relations for steady, dense granular flows

This work focuses on the mechanical response of dry granular materials under steady, simple shear conditions. In particular, the goal is to obtain a complete rheology able to describe the material behavior within the entire range of concentrations for which the flow can be considered dense. The total stress is assumed to be the linear sum of a frictional and a kinetic component. The frictional and the kinetic contributions are modeled in the context of the critical state theory and the kinetic theory of dense granular gases, respectively; in the latter, the correlated motion among the particles, which is likely to occur at high concentration, is also included. In accordance with recent findings on disordered granular packings, the frictional component of stresses is assumed to vanish when the concentration is below the random loose packing. According to this approach, four nondimensional quantities govern steady, simple shear flows: the concentration, the shear to normal stress ratio, the ratio of the time scales associated with the motion perpendicular and parallel to the flow, and the ratio between the particle stiffness and the normal stress. The present theory allows us to reproduce, in a notable way, both numerical simulations on simple shear flows of disks and physical experiments on incline flows of glass spheres taken from the literature.

A Numerical Study of Temporal Mixing Layer with Three-Dimensional Mortar Spectral Element Method

We investigate a set of well-posed interface conditions of viscous fluxes of Navier-Stokes equations, extend them to three-dimensional Navier-Stokes equations and successfully study a temporally turbulent mixing layer. The evolution of a large vortex structure and the non-dimensional statistical quantities in the mixing layer are numerically analyzed. The span-wise energy spectrum of the mixing layer exhibits wide band features, which means that the simulation generates the desired amount of small scales. These results show that too many small scales are present due to insufficient dissipation and the amount of large scales is low, indicating too much dissipation for large-scale structures.

A non-dimensional approach to diffraction phenomena in the Fresnel approximation

The paper presents a theoretical and numerical study of the diffraction problem approached in the Fresnel approximation. The problem was formulated by a non-dimensional approach, which implies the definition of two non-dimensional quantities: F and ζ. In particular the parameter F is the well-known Fresnel number, whose value was usually used to classify diffraction regime. In analyzing the numerical approach some limitations imposed by the discretization arise; in particular these limitations constrain the level of sampling to be used for the diffracting field by requiring more and more elements when the diffraction conditions become more severe — i.e. increasing the dimension of the diffracting aperture and/or decreasing the distance of the plane where the diffracted field is observed.

Investigation of the ionized gas flow adjacent to porous wall in the case when electroconductivity is a function of the longitudinal velocity gradient

This paper studies the laminar boundary layer on a body of an arbitrary shape when the ionized gas flow is planar and steady and the wall of the body within the fluid porous. The outer magnetic field is perpendicular to the fluid flow. The inner magnetic and outer electric fields are neglected. The ionized gas electroconductivity is assumed to be a function of the longitudinal velocity gradient. Using transformations, the governing boundary layer equations are brought to a general mathematical model. Based on the obtained numerical solutions in the tabular forms, the behavior of important non-dimensional quantities and characteristics of the boundary layer is graphically presented. General conclusions about the influence of certain parameters on distribution of the physical quantities in the boundary layer are drawn.

Well function curves for different geometric situations in a large-diameter well

Movement and abstraction of groundwater in the geological formations are dependent on the hydro-geological parameters of the aquifers. The purpose of any aquifer test is to determine the hydro-geological parameters. Among the basic parameters are the specific storage, permeability and leakage coefficients. The hydro-geological parameters are hidden in the field test data and their identification is possible using the available physically plausible models suitable for the prevailing field circumstances. In this context, a generalized theoretical solution for the effect of partial penetration superimposed over the full penetration on draw-down in a large-diameter well in artesian aquifer discharging at a constant rate has been presented for non-dimensional quantities describing the variable geometries of wells. The well-function curves are developed by varying the percentage amount of drilling and the percentage amount of casing lowered which then control to vary the percentage amount of open-hole or screened interval for the three categories: when the diameter of the cased interval in which the water level changes is greater than, equal to, and less than the diameter of the open interval. The skin effect and the effect of leakage are neglected. A comparison of results with the published works has also been presented. The present study is useful in such areas where wells are located either in harder or in collapsible loose formations; and a decision is required that, at the planning, construction, or development stage, as to what extent the amount of drilling be reduced, and/or an additional amount of casing be lowered within the aquifer. Also this reduces the cost of well construction and development in a specific situation.