Flow Similarity Research Articles

Similarity solution of oblique impact of wedge-shaped water column on wedged coastal structures

The initial stage of plunging wave impact obliquely on coastal structures is analysed. The problem is modelled through an oblique collision of an asymmetrical water wedge and an asymmetrical solid wedge. The gravity effect on the flow is ignored based on the assumption that the ratio of the incoming speed of the wave to the acceleration due to gravity is much larger than the time scale of interest. Similarity solution method based on the velocity potential theory is then used. The problem of this similarity flow is solved by a boundary element method through the Cauchy theorem in the complex plan. Results for the wave elevation and pressure distribution are provided, including the forces and moments, effects of different impact angles and the effects of oblique impact are investigated. In particular, negative pressure near the tip of the solid wedge is observed and its implications are discussed.

Boundary-layer equations for a power-law shear-driven flow over a plane surface of non-Newtonian fluids

We study the boundary-layer similarity flow driven over a semi-infinite flat plate by a power-law shear with asymptotic velocity profile ue(y) = βyα (y → ∞, β > 0), for fluids both Newtonian and non-Newtonian. Theoretical analysis is reported to derive a range of exponents α and amplitudes β for which similarity solutions exist. The shear stress parameter f′′(0) is determined as a function of α and β.

Numerical simulation of oblique water entry of an asymmetrical wedge

The hydrodynamic problem of a two-dimensional asymmetrical wedge entering calm water obliquely at constant speed is analyzed based on the velocity potential theory. The gravity effect on the flow is ignored based on the assumption that the ratio of the entry speed to the acceleration due to gravity is much larger than the time scale of interest. The problem of this similarity flow is solved by a boundary element method together with an analytical solution for the jet based on the shallow water approximation. Various results are provided for the wave elevation, pressure distribution and force at different deadrise angles and at different oblique entry. The effects of asymmetry and horizontal speed on these results are investigated.

A note on boundary-layer similarity flows driven by a power-law shear over a plane surface

The boundary-layer similarity flow driven over a semi-infinite flat plate by a power-law shear with asymptotic velocity profile u e ( y ) = β y α ( y → ∞ , β > 0 ) is considered. Theoretical analysis is reported to derive a range of exponents α and amplitudes β for which similarity solutions exist. The shear stress f ″ ( 0 ) parameter is determined as a function of α and β .

Power series solutions of momentum and energy boundary layer equations for a power-law shear driven flow over a semi-infinite flat plate

The boundary layer similarity flow past an impermeable flat plate, driven by a power law velocity profile U = γy α , y → ∞ is considered and power series solutions of the momentum equation, valid for all the allowed range of the parameter α, are presented. The convergence radius of the proposed solutions is estimated and a comparison with numerical solutions is reported. The boundary layer energy equation is then considered for all the wall temperature profiles that admits similarity solutions and power series solutions are given for the full range of the wall temperature profile parameter n.

Influence of Real-Fluid Properties in Modeling Decompression Wave Interacting with Ductile Fracture Propagation

The force causing the ductile fracture propagation in pipelines containing pressurized compressible fluids is given by the residual pressure of the discharging fluid, acting on the internal wall downstream of the crack tip, in the opening flap area. The value of this force is determined by the pressure profile of the rarefaction wave propagating in the pipeline. The classical treatment of this phenomenon is based on the assumption of perfect gas behaviour and similarity flow, in which conservation of entropy holds. In this context real hydrocarbon fluids are considered, and simulated by using real EOS with special focusing on two-phase flow conditions in the gas-condensate regime, assuming a homogeneous thermodynamic equilibrium model. Real-fluid properties influence the decompression wave propagation, through condensation/vaporization phenomena, sound velocity trend and frictional effects. A treatment of the thermodynamics and fluid mechanics of this problem is presented, as well as simulation results relevant to industrial applications.

Accurate estimate of disturbance amplitude variation from solution of minimal composite stability theory

Minimal composite theory (Proc. R. Soc. Lond., Ser. A, 453 2537–2549, 1979) shows that, at the lowest order in the reciprocal of the local flow Reynolds number R, the stability of a spatially developing similarity flow may be described by an ordinary differential equation in the similarity coordinate. It is, in principle, not possible to determine the dependence of the disturbance amplitude on the streamwise coordinate solely from such an ordinary differential equation. However, noting that, to O(R−2/3), the dependence of the eigenfunction on the normal coordinate is identical in both the full non-parallel and minimal composite theories, and using a method due to Gaster, we show how the streamwise variation of disturbance amplitude can be determined to O(R−1 without solving a partial differential equation, although knowledge of the partial differential operator is required. Comparison with the DNS results of Fasel and Konzelmann shows excellent agreement with the present results. Furthermore, especially in strong adverse pressure gradients, the present amplitude ratio estimates are within 3% of the full non-parallel theory, whereas the Orr–Sommerfeld results show an underestimate by 26%.

Large‐Reynolds‐Number Asymptotics of the Berman Problem

Similarity flow of a viscous fluid in a channel is considered, driven by uniform withdrawal of the fluid through the channel walls. The nonlinear ordinary differential boundary value problem that results has several branches of solutions; those of Types III, III1, and I1 are investigated here, in the limit of large wall‐suction Reynolds number. This paper gives a markedly more accurate Type III asymptotic solution than previously available, and describes the true asymptotics of the other branches for the first time. The asymptotic structure of the Type III1 solution is particularly subtle, requiring matching between seven different layers. Numerical solutions of the boundary value problem provide support for the asymptotic solutions obtained.

Numerical simulation and experimental study of water entry of a wedge in free fall motion

The hydrodynamic problem of a two-dimensional wedge entering water through free fall motion is analysed based on the velocity potential theory. The gravity effect on the flow is ignored as our interest is over a short period of time. When the tip of the wedge touches water the flow is assumed to be similar. The problem of this similarity flow is solved by a boundary element method together with an analytical solution for the jet based on the shallow water approximation. This is then used as the initial condition for the subsequent solution which is obtained by the same boundary element method in a stretched coordinate system together with the time marching technique to follow the body motion and the free surface deformation. An auxiliary function is introduced to decouple the mutual dependence of the hydrodynamic force and the body acceleration. Experimental study is also undertaken. The numerical predication is found to be in good agreement with the measured data for wedges with different weight, dead-rise angle and entry speed.

Heat transfer characteristics of a boundary-layer flow driven by a power-law shear over a semi-infinite flat plate

The hydrodynamic and thermal boundary layer similarity flows driven past a semi-infinite impermeable flat plate by a power-law shear with asymptotic velocity profile U ∞( y)= βy −1/2 ( y→∞, β>0) is considered ( y denotes the coordinate normal to the plate). Assuming that the buoyancy and viscous dissipation effects may be neglected, the special cases of an isothermal and of an adiabatic flat plate are examined both analytically and numerically.

Boundary-layer similarity flows driven by a power-law shear over a permeable plane surface

The boundary-layer similarity flow driven over a semi-infinite permeable flat plate by a power-law shear with asymptotic velocity profile U ∞(y)=βy α(y→∞,β>0) is considered in the presence of lateral suction or injection of the fluid (y denotes the coordinate normal to the plate). The analytically tractable cases α=−2/3 and α=−1/2 are examined in detail. It is shown that while for α=−2/3 the adjustment of the flow over an impermeable plate to the power-law shear is not possible, in the permeable cases the presence of suction allows for a family of boundary-layer solutions with the proper algebraic decay. The value of the skin friction corresponding to this family of solutions is given by the parameter s=9β3/(4f w ), where f w denotes the suction parameter. In the limiting case of a vanishing suction and a properly vanishing value of the parameter β (such that s=finite), this family of algebraically decaying solutions goes over into the (exponentially decaying) Glauert-jet. In the case α=−1/2, solutions showing the proper algebraic decay were found both for suction ( f w > 0) and injection ( f w <0) in the whole range −∞<f w <+∞. In this case the skin friction parameters s=2β2/3 is independent of the suction/injection parameter f w .

Analytical study of the hydrodynamics of similarity flows from thin-foil laser plasmas

The hydrodynamic expansion of plasmas produced by laser beams focused on thin-foil targets is studied using a self-similar approach. The model is useful for studying long-scale-length plasmas, produced by uniform laser irradiation, which become fully underdense during the laser pulse. An essentially qualitative analysis of the hydrodynamics system in Cartesian geometry is given. In particular, a useful expression is obtained by integration of the thermal energy equation.

Multiple Inviscid Solutions for the Flow in a Leading-Edge Vortex

To analyze the flowfield inside the vortex formed at the leading edge of a highly swept wing at an angle of attack, conical similarity solutions of the compressible Euler equations have been obtained and compared to incompressible conical similarity flow solutions. It is shown that, in contrast to the incompressible flow case, the velocity components inside the vortex core remain finite in the interior of the vortex. The compressible and inviscid flow solution exhibits vacuum conditions at the axis of the vortex core. Two different families of solutionshave been numerically obtained, one representing jetlike swirling flows olutions and the other wakelike swirling flow solutions. Considering the flow solutions in more detail reveals that, at the edge of the vortex core, the normal component of the velocity is negative (inflow) for jetlike swirling flows and positive (outflow) for wakelike swirling flows. The velocity and the vorticity vector are parallel for jetlike swirling flows and antiparallel for wakelike swirling flows. Furthermore, the azimuthal component of the vorticity has a different direction for both flow solutions. Varying the boundary conditions, for instance, the swirl number or the Mach number, at the edge of the vortex core does not change the character of the flow solutions significantly.

Boost-invariant running couplings in effective Hamiltonians

We apply a boost-invariant similarity renormalization group procedure to a light-front Hamiltonian of a scalar field phi of bare mass mu and interaction term g phi^3 in 6 dimensions using 3rd order perturbative expansion in powers of the coupling constant g. The initial Hamiltonian is regulated using momentum dependent factors that approach 1 when a cutoff parameter Delta tends to infinity. The similarity flow of corresponding effective Hamiltonians is integrated analytically and two counterterms depending on Delta are obtained in the initial Hamiltonian: a change in mu and a change of g. In addition, the interaction vertex requires a Delta-independent counterterm that contains a boost invariant function of momenta of particles participating in the interaction. The resulting effective Hamiltonians contain a running coupling constant that exhibits asymptotic freedom. The evolution of the coupling with changing width of effective Hamiltonians agrees with results obtained using Feynman diagrams and dimensional regularization when one identifies the renormalization scale with the width. The effective light-front Schroedinger equation is equally valid in a whole class of moving frames of reference including the infinite momentum frame. Therefore, the calculation described here provides an interesting pattern one can attempt to follow in the case of Hamiltonians applicable in particle physics.

Similarity renormalization, Hamiltonian flow equations, and Dyson’s intermediate representation

A general framework is presented for the renormalization of Hamiltonians via a similarity transformation. Divergences in the similarity flow equations may be handled with dimensional regularization in this approach, and the resulting effective Hamiltonian is finite since states well-separated in energy are uncoupled. Specific schemes developed several years ago by Glazek and Wilson and contemporaneously by Wegner correspond to particular choices within this framework, and the relative merits of such choices are discussed from this vantage point. It is shown that a scheme for the transformation of Hamiltonians introduced by Dyson in the early 1950's also corresponds to a particular choice within the similarity renormalization framework, and it is argued that Dyson's scheme is preferable to the others for ease of computation. As an example, it is shown how a logarithmically confining potential arises simply at second order in light-front QCD within Dyson's scheme, a result found previously for other similarity renormalization schemes. Steps toward higher order and nonperturbative calculations are outlined. In particular, a set of equations analogous to Dyson-Schwinger equations is developed.

Properties of similarity solutions for flows in turbulent vortex cores

A class of similarity solutions of the equations for turbulent vortex cores matching an external inviscid similarity flow with a power law of circumferential velocity variationv-r −m near the rotation axis and constant Bernoulli function is considered. Solutions are found to exist only in a certain range of the indexm of the exponential. For each suchm there are two solutions.

Characteristics of endwall and sidewall boundary layers in a rotating cylinder with a differentially rotating endwall

The flow in a rotating cylinder driven by the differential rotation of its top endwall is studied by numerically solving the time-dependent axisymmetric Navier–Stokes equations. When the differential rotation is small, the flow is well described in terms of similarity solutions of individual rotating disks of infinite radius. For larger differential rotations, whether the top is co-rotating or counter-rotating results in qualitatively distinct behaviour. For counter-rotation, the boundary layer on the top endwall separates, forming a free shear layer and this results in a global coupling between the boundary layer flows on the various walls and a global departure from the similarity flows. At large Reynolds numbers, this shear layer becomes unstable. For a co-rotating top, there is a qualitative change in the flow depending on whether the top rotates faster or slower than the rest of the cylinder. When the top rotates faster, so does the bulk of the interior fluid, and the sidewall boundary layer region where the fluid adjusts to the slower rotation rate of the cylinder is centrifugally unstable. The secondary induced meridional flow is also potentially unstable in this region. This is manifested by the inflectional radial profiles of the vertical velocity and azimuthal vorticity in this region. At large Reynolds numbers, the instability of the sidewall layer results in roll waves propagating downwards.

Equations of state compatible with similarity flows

An analysis of the role of the equation of state as a regulator to the existence of ``test'' and self-gravitating relativistic self-similar flows is presented. Contrary to the prevailing usage of $P=k\ensuremath{\rho},$ $0&lt;~k&lt;~1$ as the only physically realistic equation of state compatible with relativistic self-similarity, it is explicitly demonstrated that this is in fact not necessary. We show that the similarity ansatz allows flows which obey a polytropic equation of state. Explicit spherical self-similar flows are presented. It is also argued that a large class of equations of state do not exclude relativistic self-similar flows. Furthermore it is shown that the coupling of the Einstein tensor to a self-similar flow results in the elimination of self-gravitating self-similar flows obeying, for instance, a polytropic equation of state, though room is still left for other nonisentropic families of equation of state.

Boundary layer similarity flow driven by power-law shear

Similarity solution of the Prandtl boundary layer equations describing wallbounded flows and symmetric free-shear flows driven by rotational velocitiesU(y)=βyα are determined for a range of exponents α and amplitudes β. Asymptotic analysis of the equations shows that for α −2/3 the shear stressf″(0) parameter is determined as a function of α and β. Symmetric free-shear flow solutions become singular as α→α0 = −1/2 and no solutions are found in the range −1 −1/2 the centerline velocityf′(0) is determined as a function of α and β. An asymptotic analysis of the singular behavior of these two problems as α→α0, given in a separate Appendix, shows excellent comparison with the numerical results. Similarity solutions at the critical values α0 have exponential decay in the far field and correspond to the Glauert wall jet for wall-bounded flow and to the Schlichting/Bickley planar jet for symmetric free-shear flow.

Hydrodynamics of Ultra-Relativistic Heavy Ion Collisions. Considerations of Boost Non-invariance and Stopping

We investigate the features of boost-invariant hydrodynamics of Bjorken, the boost-non-invariant hydrodynamics of Srivastava et al. and the Landau hydrodynamics corresponding to the so-called stopping scenario for the ultra relativistic collision of heavy nuclei. The chemical potential has been assumed to be small, and a (1 + 1)-dimensional expansion is considered. The fluid rapidity densities for the three cases are found to be quite distinct in nature. It is shown that a rapidity shift parameter can be defined which measures the deviation of the flow from the so-called similarity flow. It is found to be dramatically different for the three hydrodynamic solutions and can be used to characterise the flow. The cooling laws for the three solutions are obtained and they reveal that the cooling is fastest for the Landau solution and slowest for the Bjorken solution. It is also seen that the initial temperatures likely to be attained at a given incident energy are largest for the Landau solution and smallest for the Bjorken solution. A number of alternative general solutions are also discussed.