Transverse Curvature Effects Research Articles

Theoretical analysis of radiation effects in laminar flow over a von-Kármán body of revolution

An analysis presents the impact of thermal radiation on the laminar boundary layer established due to the motion of a von-Kármán surface. The reason for choosing such a shape is its appearance in many aeronautical vehicles, especially the F-16 fighter jet nose cone. However, the transverse curvature effects are also observable in many other shapes of bodies of revolution. But considering such a practical body of revolution provides a chance to quantify the transport phenomena happening on sophisticated aerospace vehicles. As a result, a correct understanding of the momentum and the thermal process in actual situations can be achieved. Power-law form of wall temperature is assumed to cover a variety of non-isothermal wall conditions. The typical body contour of a von-Kármán surface does not allow the scaling symmetry in the longitudinal direction; thus making the flow of non-similar in nature. An efficient numerical scheme (Keller–Box) with the second order accuracy is used for the solution purposes. The results are found to be satisfactory regarding the previous published work for moving cylinder cases. This ensures the validity of the tabulated data for the von-Kármán surface case. Through numerous tables and graphs, the impacts of thermal radiation parameter, wall temperature exponent, and transverse curvature parameter have been highlighted and thoroughly analyzed. Once the radiation parameter is increased, it is noticed that the rate of heat transfer increases and the thermal thickness of the boundary layer grows, whereas the reverse behavior is seen when the wall temperature exponent is increased. Furthermore, it has been found that increasing the surface transverse curvature increases the thermal thickness of the boundary layer and the rate of heat transfer. To the considered body shape, the reported data are expected to serve as a good source for the development of approximate methods concerning complex flow geometries involving transverse curvature effects.

Boundary layer measurements of the flow along a streamwise-oriented circular cylinder using particle tracking velocimetry

Particle tracking velocimetry (PTV) measurements of an axisymmetric turbulent boundary layer (ATBL) around a long circular cylinder in axial flow were carried out in a large-scale water tunnel at different velocities and at different measurement positions along the cylinder. A relatively thick ATBL developed with ratios of δ/a between 4.6 and 8.1 and a/δν between 313 and 1027, where δ is the boundary layer thickness, δν the viscous length scale and a the cylinder radius. At ratios of δ/a above unity and a/δν above 250, transverse curvature effects are known to influence the boundary layer in the outer region. This was observed in the measured mean velocity profiles as the profiles fall below the planar logarithmic law of the wall. Shape factors close to unity with a rather full velocity profile were determined that explain the relatively high skin friction coefficient known to be present for such ATBL flows. Additionally, measurements of the turbulence intensity and Reynolds stress profiles are presented. The non-intrusive PTV measurement approach allowed to gather near-wall velocity data down to a wall-distance below 10 wall-units. The present study provides valuable information regarding ATBL flows along streamwise-oriented cylinders in the flow regime indicated above by the ranges of δ/a and a/δν. In particular, it fills a gap in the literature as far as experimentally obtained higher-order statistical quantities are concerned, such as root-mean-square velocity fluctuations and Reynolds stresses.

Combined effects of surface sharpness and transverse curvature on thermal transport in separating flow region

Heat transfer analysis in a retarded laminar boundary layer over bodies of revolution with a variable radius subject to retarded surface temperature is presented. This study focuses to develop numerical data for local Nusselt number N ux under the influence of retarded free stream velocity and decreasing surface temperature along with body contour and surface transverse curvature. With the use of such retarded velocities and body contour the similarity of flow phenomenon does not sustain, and the problem becomes non-similar in nature which has been solved using an efficient numerical scheme, commonly known as Keller-Box. The effects of pertinent parameters such as exponent of free stream velocity m, exponent of surface temperature q, body contour n, and surface transverse curvature k on temperature profile, rate of heat transfer at the surface, and energy thickness of thermal boundary layer are presented through various graphs and tables. The rate of heat transfer and the energy thickness of the thermal boundary layer are observed to decrease with increasing levels of retardation. Furthermore, the rate of heat transfer decreases as the body contour parameter increases, whereas the surface transverse curvature parameter exhibits the opposite trend.

Numerical investigation of the effect of convex transverse curvature and concave grooves on the turbulent boundary layer along a cylinder in axial flow

Axisymmetric turbulent boundary layers that develop around streamwise oriented long cylinder-like objects can be found in many applications, such as towed array sonars or marine seismic streamers. In many of these applications, turbulent fluctuations within the boundary layer flow can have a negative impact compared with laminar flow conditions. The aim of the present work is to design a surface modification that influences the turbulent boundary layer around a cylinder in axial flow in order to reduce turbulent fluctuations. To design the surface we consider recent findings regarding the turbulence damping effects of groove-like surface structures and combine these insights with the effect of convex transverse curvature on turbulence. We use large-eddy simulations to investigate the flow around a cylinder of modified design and around a reference circular cylinder. Both flows have a radius-based Reynolds number of Rea≈1.23⋅104. The modified design leads to a 20 % decrease in the average wall shear stress and results in local reductions in the turbulent intensities, Reynolds stress, the temporal velocity spectrum, and the turbulent dissipation rate. The analysis within the anisotropy-invariant space reveals a tendency towards flow relaminarization. However, the new design has no effect on turbulent pressure fluctuations. We provide suggestions on how to further improve the surface design to achieve even greater flow stabilization.

Global stability analysis of axisymmetric boundary layer on a slender circular cone with the streamwise adverse pressure gradient

This paper presents a global stability analysis of the boundary layer developed on a slender circular cone. The base flow direction is towards the apex of a cone, and the pressure gradient is positive (adverse) in the flow direction. The decelerating base flow is non-parallel and non-similar. The increased semi-cone angle increases the adverse pressure gradient in the flow direction. For a given semi-cone angle, transverse curvature increases in the flow direction. However, increased semi-cone angle reduces transverse curvature at any streamwise location. The Reynolds number of the flow is defined based on the displacement thickness at the computational domain’s inlet. The radius of the cone reduces in the flow direction, which results in the increased transverse curvature. The governing stability equations are derived in the spherical coordinates and discretized using the Chebyshev Spectral collocation method. The discretized equations, along with homogeneous boundary conditions, form a general eigenvalue problem, and it is solved using Arnoldi’s iterative algorithm. The global temporal modes have been computed for small semi-cone angles α=2°, 4°, and 6°, azimuthal wave-numbers N=0, 1, 2, and 3 and Re=283, 416, and 610. Thus, the state of base flow is laminar at the inlet (line pq) of the domain pqrs. All the global modes computed within the range of parameters are found temporally stable. Further, the global modes are found less stable at higher semi-cone angles (α) due to the streamwise adverse pressure gradient. The effect of transverse curvature has been found significant at higher semi-cone angles. For a given Re and α, the global modes are found least stable for the helical mode N=1 and most stable for N=3. The eigenmodes’ two-dimensional spatial structure suggests that flow is spatially unstable as the disturbances grow in the streamwise direction.

TVC effects on flow separation on slender cylinders

Occurrence of flow separation within the boundary-layer is primarily attributed to the growth of adverse pres-sure gradient within the boundary-layer. Such a separating boundary-layer can, however, be made to stay attached to the bounding surface due to various remedies where the surface transverse curvature (TVC) is one of them while the body contour is another. Assistive role of the surface transverse curvature is utilized in delaying the boundary-layer separation in a retarded flow past the slender bodies by assuming different body configurations. Axially-symmetric bodies of revolution of different shapes having a varying cross section in x, following the power-law form, have been considered in this study. It is shown that the point of separation is strongly dependent upon the surface curvature parameter k and the body-shape index n, which in fact control the flow separation. This fact ensures that an increase in the surface curvature parameter k and the body-shape index n leads to an increase in the wall shear stress which consequently delays the flow separation. Significant delay in flow separation is observed on cylinders having sufficiently large transverse curvature. Percent increase in the separation length corresponding to various values of the curvature parameter and the body-shape index has been calculated and reported in the form of Tables.

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

Natural Convection and Separation Points of a Non-Newtonian Fluid Along a Rotating Round-Nosed Body

The purpose of the present study is to present a numerical solution for the natural-convection flow of Ostwald/de Waele-type power-law non-Newtonian fluid along the surface of a rotating axisymmetric round-nosed body. For computational purpose, a rotating hemisphere is used as the case study to examine the effects of transverse curvature on heat transfer mechanism. The numerical scheme is applied after converting the dimensionless system of equations into primitive variable formulations. Implicit finite difference method is used to integrate the equations numerically. The numerical simulations performed are valid for shear-thickening fluids with a wide range of Prandtl numbers [i.e., ]. A detailed discussion is provided to understand the effects of buoyant forces and power-law exponent on the rate of heat transfer and skin-friction coefficient. A comparison of the present numerical results for different values of the buoyancy ratio parameter with other published data is shown in graphical form. The flow separation occurs when a portion of the boundary layer closest to the wall or leading edge reverses in flow direction. It is observed from our results that an increase in the power-law index and Prandtl number leads to an increase in the friction factor, as well as the rate of heat transfer.

Highly-resolved LES of turbulent convective flow along a PWR rod bundle

A detailed numerical analysis of the turbulent convective flow along the heated rods of an idealized Pressurized Water Reactor (PWR) sub-channel is investigated using the CFD code TransAT. The flow is pretty much similar to circular pipe flow. Turbulent effects are predicted using highly-resolved Large-Eddy Simulation (LES) with a grid resolution of up to 6 million cells, resolving the viscous-affected layer. The sub-grid scale (SGS) viscosity produced by the model is indeed found to be of marginal effect for the grid and Reynolds number employed. Only first-order turbulence statistics are presented here. The results are discussed in detail, in particular key features specific to rod bundles, including low-Re effects in the narrow gap zone and the strong secondary flow motion, which is shown to exceed the turbulence counterpart (through the shear stress) near the wall. The secondary-flow motion induced by the mean flow is shown to be responsible for a large portion of the wall-to-flow heat transfer. The comparison of the LES results with existing DNS of pipe flow shows a very good agreement as to first-order statistics; higher-order statistics (including energy budgets) of the fluctuating fields have not been explored. A data basis has been generated for turbulence model comparison. Like in turbulent pipe flow, a physical explanation for the observed differences can be routed in the transverse curvature effects of the bundle geometry.

Global stability analysis of axisymmetric boundary layer over a circular cone

A linear global stability analysis is performed for an axisymmetric boundary layer formed on a circular cone. The combined effect of transverse curvature and favorable pressure gradient has been studied. The increase in semicone angle makes the flow temporally stable and convectively unstable.

RANS Based Computational Fluid Dynamics Simulation of Fully Developed Turbulent Newtonian Flow in Concentric Annuli

A computational fluid dynamics (CFD) simulation is performed via ansys–CFX for a fully developed turbulent flow in concentric annuli with two radius ratios (R1/R2 = 0.4 and 0.5) at three Reynolds numbers (Re = 8900, 26,600, and 38,700) in terms of the hydraulic diameter D and the bulk velocity Ub. Near-wall turbulence structures close to the inner and outer walls are characterized by analyzing the first-order and second-order statistics. Effects of transverse curvature and the Reynolds number on development of the turbulence structures are emphasized. This study demonstrates the ability to predict the asymmetric features of turbulent flows in annular pipes by using the Reynolds-Averaged Navier–Stokes (RANS) model. Estimation of viscous dissipation in the flow using a RANS model is compared with direct numerical simulation (DNS) results for the first time.

Viscous dissipation and thermal radiation effects on the magnetohydrodynamic (MHD) flow and heat transfer over a stretching slender cylinder

An axisymmetric magnetohydrodynamic (MHD) boundary layer flow and heat transfer of a fluid over a slender cylinder are investigated numerically. The effects of viscous dissipation, thermal radiation, and surface transverse curvature are taken into account in the simulations. For this purpose, the governing partial differential equations are transformed to ordinary differential equations by using appropriate similarity transformations. The resultant ordinary differential equations along with appropriate boundary conditions are solved by the fourth-order Runge–Kutta method combined with the shooting technique. The effects of various parameters on the velocity and temperature profiles, local skin friction coefficient, and Nusselt number are analyzed.

The transverse curvature effects of non-Newtonian flow past a slender parabolic body of revolution

The Levy Lee transformations are modified with the aim of transforming nonlinear partial differential equations of boundary value type governing the steady two dimensional incompressible boundary layer flows of non-Newtonian power-law fluids past a slender parabolic body of revolution into nonlinear ordinary differential equation with appropriate boundary conditions. The transverse curvature effect of the present flow situation is studied theoretically using group-theoretic initial value technique. It is concluded that the slope of the velocity profile is greatly influenced by transverse curvature and suction or blowing parameters.

Strong Elongation Versus a Large Transverse Curvature in the Problems of High-Frequency Diffraction

The effects of the high transverse curvature and of the strong elongation on the high-frequency diffraction are studied on the examples of the perfectly conducting elliptic cylinders and spheroids with high aspect ratio.

MHD mixed convection of a viscous dissipating fluid about a vertical slender cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a heated/cooled vertical slender cylinder is studied numerically, taking into account the effects of transverse curvature and viscous dissipation. A uniform magnetic field is applied perpendicular to the cylinder. The resulting governing equations are transformed into the nonsimilar boundary layer equations and solved using the Keller box method. The velocity and temperature profiles as well as the local skin friction and the local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, and the Eckert number. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase, increasing the Richardson number and the magnetic parameter Mn or decreasing the Eckert number.

Axisymmetric magneto-hydrodynamic (MHD) flow and heat transfer at a non-isothermal stretching cylinder

An investigation is made to study the effects of transverse curvature and the temperature dependent thermal conductivity on the magneto-hydrodynamic (MHD) axisymmetric flow and heat transfer characteristics of a viscous incompressible fluid induced by a non-isothermal stretching cylinder in the presence of internal heat generation/absorption. It is assumed that the cylinder is stretched in the axial direction with a linear velocity and the surface temperature of the cylinder is subjected to vary non-isothermally. Here the thermal conductivity is assumed to vary linearly with temperature. Using a similarity transformation, the governing system of partial differential equations is first transformed into coupled non-linear ordinary differential equations with variable coefficients. The resulting intricate non-linear boundary value problem is solved numerically by a second order finite difference scheme for different values of the pertinent parameters for two cases: (i) the prescribed surface temperature (PST case) and (ii) the prescribed heat flux (PHF case). Numerical results are obtained for two different cases namely, zero and non-zero values of the curvature parameter to get the effects on the velocity and temperature fields. The combined effects of the curvature parameter and the thermal conductivity parameter are examined. The physical significances of the numerical results are presented for several limiting cases.

Secondary Instabilities in Incompressible Axisymmetric Boundary Layers: Effect of Transverse Curvature

The secondary instability of the incompressible boundary layer in the axial flow past a cylinder is studied. The laminar flow is shown to be always stable at high transverse curvatures to secondary disturbances. Because the primary mode is stable as well, (Tutty et al., 2002, “Boundary Layer Flow on a Long Thin Cylinder,”. Phys. Fluids, 14(2), pp. 628–637), this implies that the boundary layer on a thin long cylinder may undergo transition to turbulence by means very different from that on a flat plate. The azimuthal wavenumber of the least stable secondary modes (m±) are related to that of the primary (n) by m+ = 2n and m− = −n. The base flow is shown to be inviscidly stable at any curvature.

Free convection from a truncated cone subject to constant wall heat flux in a micropolar fluid

The paper studies the problem of free convection about a vertical frustum of a cone in a micropolar fluid. It is assumed that the flow is laminar, steady and the wall is subjected to a constant heat flux and the angle of the frustum of the cone is large enough so that the transverse curvature effects are negligible. Under these assumptions, the governing boundary layer equations subject to appropriate boundary conditions are transformed into a set of equations of parabolic type, that are solved using the local non-similarity method. The space of parameters contains the Prandtl number Pr, the micropolar parameter Δ and the microrotation parameter n. Numerical solutions are obtained by varying Pr from 6.7 to 100, Δ from 0 (Newtonian fluid) to 2 and considering two values of n with physical significance (0 and 0.5). Flow and heat transfer characteristics are determined and are shown in graphs. The results are discussed and compared at some extent with those reported by the present author in a previous study (Postelnicu in Int. J. Eng. Sci. 44:672–682, 2006) on the isothermal case.

Free convection about a vertical frustum of a cone in a micropolar fluid

The paper studies the problem of free convection about a vertical frustum of a cone in a micropolar fluid. It is assumed that the flow is steady, and the surface temperature of the frustum of the cone is constant. Another assumption is that the angles of the frustum of the cone are large enough so that the transverse curvature effects are negligible. Under these assumptions, the governing boundary layer equations subjected to appropriate boundary conditions are first written in a non-dimensional form. These equations are then transformed into a set of non-similar partial differential equations of parabolic type, which is amenable to a direct numerical solution, using a very efficient method known as Keller-box method. Numerical solutions are obtained for a range values of the micropolar parameter Δ varying between Δ = 0 (Newtonian fluid) to Δ = 2 and Prandtl number Pr is varied from 0.1 to 10. Flow and heat transfer characteristics are determined and are given in tables and also shown on graphs. The obtained results are also compared with those known from the open literature and it is found that they are in excellent agreement.

Effect of non-uniform slot injection (suction) on a forced flow over a slender cylinder

An analysis is performed to obtain the non-similar solution of a steady laminar forced convection boundary layer flow over a horizontal slender cylinder including the effect of non-uniform slot injection (suction). The effects of transverse curvature and viscous dissipation are also included in the analysis. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form using suitable transformations and the resulting system of nonlinear coupled partial differential equations is then solved by an implicit finite difference scheme in combination with the quasilinearization technique. Numerical results for the effect of non-uniform slot injection (suction) on skin friction coefficient and heat transfer rate are presented. The effects of transverse curvature, viscous dissipation and Prandtl number on velocity and temperature profiles and skin friction and heat transfer coefficients are also reported.