Parabolic Body Research Articles

Hydrodynamic and heat transfer characteristics of laminar flow past a parabolic cylinder with constant heat flux

Steady, two-dimensional, symmetric, laminar and incompressible flow past parabolic bodies in a uniform stream with constant heat flux is investigated numerically. The full Navier–Stokes and energy equations in parabolic coordinates with stream function, vorticity and temperature as dependent variables were solved. These equations were solved using a second order accurate finite difference scheme on a non-uniform grid. The leading edge region was part of the solution domain. Wide range of Reynolds number (based on the nose radius of curvature) was covered for different values of Prandtl number. The flow past a semi-infinite flat plate was obtained when Reynolds number is set equal to zero. Results are presented for pressure and temperature distributions. Also local and average skin friction and Nusselt number distributions are presented. The effect of both Reynolds number and Prandtl number on the local and average Nusselt number is also presented.

Electromagnetic radiation of antennas in the presence of an arbitrarily shaped dielectric object: Green dyadics and their applications

Although numerical solutions to the electromagnetic scattering by an arbitrarily shaped object have been obtained using Waterman's (1971) T-matrix method (TMM), the general electromagnetic radiation due to an antenna of a three-dimensional (3-D) current distribution in the presence of an arbitrarily shaped object has not been well considered. In this paper, the technique of surface integral equations has been employed; and as a result, a terse and analytical representation of the dyadic Green's functions (DGFs) in the presence of an arbitrarily shaped dielectric object is obtained for the antenna radiation. In a form similar to that associated with the electromagnetic radiation in the presence of a dielectric sphere, the DGFs inside and outside of the object of arbitrary shape are expanded in terms of spherical vector wave functions. However, their coefficients are no longer decoupled due to the arbitrary surface of a 3-D object. The coupled coefficients are then determined using the surface integral equation approach, in a fashion similar to that in the T-matrix method. To confirm the applicability and correctness of the approach in this paper a dielectric sphere, as a special case, is utilized as an illustration. It is found that exactly the same expressions as in the rigorous analysis for the inner and outer spherical regions of the object are obtained using the different approaches. As applications of the approach in this paper, radiation problems of an electric dipole in the presence of superspheroids and rotational parabolic bodies are solved.

Numerical solutions of the Navier‐Stokes and energy equations for laminar incompressible flow past parabolic bodies

Numerical solutions are presented for steady two‐dimensional symmetric flow past parabolic bodies in a uniform stream parallel to its axis. For this study, the full Navier‐Stokes equations and energy equation in parabolic coordinates were solved. A second order accurate finite difference scheme on a non‐uniform grid was used. The solution domain does not exclude the leading edge region as it is usually done with boundary layer flows. A wide range of Reynolds number (Re) is studied for different values of Prandtl number (Pr). It is found that the average Nusselt number (Nu) increases as (Pr) increases meanwhile, (Nu) decreases with the increase in (Re).

Boundary layer receptivity to free-stream sound on parabolic bodies

We use a numerical approach to study the receptivity of the boundary layer flow over a slender body with a leading edge of finite radius of curvature to small streamwise velocity fluctuations of a given frequency. The body of interest is a parabola in order to exclude jumps in curvature, which are known sites of receptivity and which occur on elliptic leading edges matched to finite-thickness at plates. The infinitesimally thin flat plate is the limiting solution for the parabola as the nose radius of curvature goes to zero. The formulation of the problem allows the two-dimensional unsteady Navier–Stokes equations in stream function and vorticity form to be converted to two steady systems of equations describing the basic (nonlinear) flow and the perturbation (linear) flow. The results for the basic flow are in excellent agreement with those in the literature. As expected, the perturbation flow was found to be a combination of an unsteady Stokes flow and Orr–Sommerfeld modes. To separate these, the unsteady Stokes flow was solved separately and subtracted from the total perturbation flow. We found agreement with the streamwise wavelengths and locations of Branches I and II of the linear stability neutral growth curve for Tollmien–Schlichting waves. The results showed an increase in the leading-edge receptivity with decreasing nose radius, with the maximum occurring for an infinitely sharp flat plate. The receptivity coefficient was also found to increase with angle of attack. These results were in qualitative agreement with the asymptotic analysis of Hammerton & Kerschen (1992). Good quantitative agreement was also found with the recent numerical results of Fuciarelli (1997), and the experimental results of Saric, Wei & Rasmussen (1994).

Dendrite Tip-Shape Characteristics

ABSTRACTThe assumption that dendrite tips are parabolic bodies of revolution pervades many of the theories and experiments addressing dendritic growth. This assumption, while reasonable, is known to become less valid as regions of interest further from the tip of the dendrite are considered. Experimental measurements were made on pure succinonitrile dendrites at several super coolings. The equation that describes the dendrite tip profile is extended from a second order polynomial (paraboloidal) form to one that includes higher-order terms. The deviation of a dendrite tip from a parabolic body of revolution can be characterized by a parameter obtained from the coefficient of the fourth-order term describing the profile. This dimensionless parameter, Q, is found to be a function of the profile orientation only, independent of supercooling.

Newtonian and hypersonic flows over oscillating bodies of revolution—II Parabolic bodies

Recently developed unsteady hypersonic and Newtonian perturbation theories for flow past slender pointed-nose bodies of revolution pitching about any pivot position are further studied in this paper. In Part I, circular cones were considered; here we consider bodies described by second degree equations. It is found that the circumferential speed is singular at body surface for both circular cones and second degree polynomials for pivot positions other than the vertex and is regular for pitching oscillations about the vertex. Also it is found that the in-phase component of the unsteady shock wave does not satisfy the shock attachment condition, but both the in-phase and the out-of-phase components of the stability derivative are regular for all pivot positions. The surface(convex)curvature is found to decrease the out-of-phase component of the stability derivative, but the oscillations remain dynamically stable.

Hypersonic aircraft and inlet configurations derived from axisymmetric flowfields

Waveriders and inlets derived from the stream surfaces of axisymmetric Newtonian flows past slender pointed-nose parabolic bodies at high Mach numbers are constructed. The flowfield of the waverider including pressure distribution, velocity components, and the attached shock wave are all known in closed-form. Thus, the variation of lift and drag forces with the various parameters and the mass flow rate through the inlet are found. It is shown that adding some longitudinal curvature to the surfaces of the cone-derived waverider, considerably increases the lift-to-drag ratio but decreases the lifting force.

Viscous Flowfield Solutions about a Finite Parabolic Body

AHYBRID computational technique that splits the flowfield into inviscid and viscous regions is used to investigate the complete flowfield about axisymmetric parabolic blunt bodies in a supersonic stream. The solutions are carried out on the CDC CYBER-203 computer which, with its extensive memory, allows for the use of a large number of finite-differ ence mesh points, allowing resolution of important flowfield features. Contents Analysis of the supersonic flow about blunt bodies has in the past been restricted by limited computational resources to flow over the forebody. This has led to some computational difficulties when the sonic line has moved outside of the computational domain; and, has left unresolved, the effect of base flow on the total drag of a blunt body. We have extended the hybrid computational technique described in Ref. 1 to investigate the complete flowfield about axisymmetric parabolic blunt bodies in a supersonic stream. The solutions are carried out on the CDC CYBER-203 vector computer, which with its extensive memory, allows for the use of a large number of finite-differ ence mesh points, allowing resolution of important flowfield features. The physical coordinate system used in the description of the body and flowfield is the orthogonal parabolic system used in Ref. 2 and is formed by the intersection of curves belonging to two cofocal parabolas. The parametric equations for the Cartesian coordinates in this system are x=lA (£2 — 772) and y = £r\. As seen in Fig. 1, the axisymmetric body surface is formed by rotating the coordinate lines about the X axis. As described in Ref. 1, the flowfield is split into viscous and inviscird regions that are coupled along the line 77 = rjc, Fig. 1. The computational plane, in £ and 77, is rectangular with the physical boundaries corresponding to the simple rectangular boundaries of the computational plane. Coordinate stretching/compression is applied both in the forebody and afterbody regions to resolve important flowfield features. Because the shock-fitting technique of Ref. 1 is used, the flow near the shock is considered to be inviscid and is governed by the conservative, compressible, time-dependent Euler equations. The remaining flowfield is governed by the full compressible, laminar, time-dependent Navier-Stokes equations in conservative form. The governing equations are cast in the form

Two-phase flow and heat transfer in boundary layer of a thin body of revolution

The effect of transverse curvature of a body and of the amount of injection on flow and heat transfer in a two-phase boundary layer of a thin parabolic body of revolution is investigated.

An analysis of second-order effects on laminar boundary-layer flow

Treated in the present paper are the second-order effects, i.e. the effect of displacement thickness, longitudinal curvature, external shear and slip, on the two-dimensional laminar boundary-layer flow of incompressible fluid. The analysis is developed in terms of the stream-function co-ordinates proposed for the analysis of the rotational flow by the senior author previously. An inverse problem is set in the co-ordinates plane, and, within the framework of the second-order approximation, the problem is reduced to solving a parabolic partial differential equation for the total head. An implicit finite-difference scheme is then devised to solve the equation, and the practical computations are carried out for the flow along a flat plate or the one around a parabolic body, by making use of an electronic computer.

Stability of Finite-Difference Equation for the Transient Response of a Flat Plate

ment is obtained throughout the supersonic range of M = 1.0 to 4.0. Figure 2 shows the effect of varying the over-all fineness ratio for ogive-cylinder combinations at freestream Mach numbers from 1.5 to 3.0. The effect of ogive-to-cylinder length ratio is small as compared to the effect of the over-all fineness ratio, which affects average skin friction greatly below an over-all fineness ratio of approximately 7. Figure 3 compares the effects of various body shapes on local skin-friction coefficient, for a typical freestream Mach number of 2.0. The ogive cylinder, cone, and parabolic body of revolution show somewhat the same general skin-friction distribution along their forebodies where pressure gradients are favorable (dp/dx < 0). The effects of the adverse pressure gradients on the local skin-friction distribution are most pronounced on the parabolic body of revolution which has a rapidly increasing pressure near the aft end. This creates a rapidly falling skin-friction coefficient that approaches the point of separation. The calculation procedure appears to be capable of estimating the separation point, in the case of adverse pressure gradient, although no attempt to verify its accuracy is made.

Supersonic flow past a bluff body with a detached shock Part I Two-dimensional body

The flow at high Mach number past a body with a rounded nose is considered. Viscosity and heat conduction are neglected, and the body is assumed to be two-dimensional and symmetrical about an axis parallel to the incident stream.An exact solution is first obtained in the case λ → 1, M → ∞, where λ is the adiabatic index and M is the Mach number. This solution is then used as the basis of a double expansion in δ = (λ - 1)/(λ + 1) and M−2, after the exact solution has been modified to make the series expansion converge near the body. The expansion is carried out as far as the terms of order (δ + M−2)2.The results are displayed for various values of δ and M−2; typical results are as follows. With M−2 = 0, δ = 1/6 (λ = 1.4), and for a parabolic body having unit radius of curvature at the nose, the shock is approximately a parabola with radius of curvature 1.822 at the nose. The distance between the body and the shock along the axis of symmetry is 0.323, and the height of the sonic point from this axis is 0.744 both on the shock and on the body. The actual pressure distribution on the body is shown in figure 4, and agrees well with experiment. The pressure falls to zero at a distance 0.86 downstream from the nose of the body, measured along the axis of symmetry. On the assumption that the pressure remains negligible beyond this point, the total drag is 1.39 ρV2, where ρ0 is the density andV is the velocity of the incident stream.