Similarity Form Research Articles

An Exact Similarity Solution for Spherical Shocks in a Relaxing Gas

In this paper, we have found an exact similarity solution of the point explosion problem concerning spherical symmetric flows in a relaxing gas. In contrast to situations without relaxation, the inclusion of relaxation effects implies constraint conditions. We assume the solution of the problem in a similarity form which transform the given PDEs with jump conditions to a set of ODEs with boundary conditions. Then, following McVittie technique, a general solution of particle velocity, density and pressure have been written into a similarity form. This similarity form of velocity, density and pressure together with set of ODEs with boundary conditions produces the complete solution. These solutions are quite general and the method is quite different with the conventional ones.

Equilibrium similarity solution of the turbulent transport equation along the centreline of a round jet

A novel similarity-based form is derived of the transport equation for the second-order velocity structure function of$\langle ({\it\delta}q)^{2}\rangle$along the centreline of a round turbulent jet using an equilibrium similarity analysis. This self-similar equation has the advantage of requiring less extensive measurements to calculate the inhomogeneous (decay and production) terms of the transport equation. It is suggested that the normalised third-order structure function can be uniquely determined when the normalised second-order structure function, the power-law exponent of$\langle q^{2}\rangle$and the decay rate constants of$\langle u^{2}\rangle$and$\langle v^{2}\rangle$are available. In addition, the current analysis demonstrates that the assumption of similarity, combined with an inverse relation between the mean velocity$U$and the streamwise distance$x-x_{0}$from the virtual origin (i.e. $U\propto (x-x_{0})^{-1}$), is sufficient to predict a power-law decay for the turbulence kinetic energy ($\langle q^{2}\rangle \propto (x-x_{0})^{m}$), rather than requiring a power-law decay ($m=-2$) as an additionalad hocassumption. On the basis of the current analysis, it is suggested that the mean kinetic energy dissipation rate,$\langle {\it\epsilon}\rangle$, varies as$(x-x_{0})^{m-2}$. These theoretical results are tested against new experimental data obtained along the centreline of a round turbulent jet as well as previously published data on round jets for$11\,000\leqslant \mathit{Re}_{D}\leqslant 184\,000$over the range$10\leqslant x/D\leqslant 90$. For the present experiments,$\langle q^{2}\rangle$exhibits power-law behaviour with$m=-1.83$. The validity of this solution is confirmed using other experimental data where$\langle q^{2}\rangle$follows a power law with$-1.89\leqslant m\leqslant -1.78$. The present similarity form of the transport equation for$\langle ({\it\delta}q)^{2}\rangle$is also shown to be closely satisfied by the experimental data.

Rise dynamics of unsteady laboratory jets with implications for volcanic plumes

Geophysical observations of discrete volcanic eruptions suggest their eruption rates vary rapidly in time. To learn how such variations may impact the initial stages of plume rise, we conducted a laboratory study of neutrally-buoyant jets generated by unsteady source conditions analogous to the volcanic case. Turbulent jets were generated by quickly injecting a finite volume of water into a large volume of still water. The mass injection rates evolved over time with a Gaussian-like history, producing jets with peak Reynolds numbers ranging from 104 to 105, consistent with values estimated for small, discrete eruptions. Except during very early and late times, jet heights show a logarithmic dependence on time; this trend contrasts with the power law dependence for jets produced by steady-state or instantaneous discharge conditions. We show that this logarithmic dependence is the similarity form appropriate for impulsive releases from a time-varying source, and found characteristic length and time scales that consolidate the non-dimensional jet heights, as functions of non-dimensional times, from a range of experimental conditions onto a single trend. The rise of unsteady volcanic plume fronts from short-duration eruptions (Mori and Burton, 2009; Patrick, 2007) show the same trend as that observed in the laboratory. Variations in mass eruption rate strongly influence the initial phases of plume rise and may impact related processes such as mixing, entrainment and eruption column collapse. Consequently, source unsteadiness must be accounted for in physical plume models before they will reliably estimate trajectory, dilution, and stability for volcanic plumes from discrete eruptions.

MHD Stagnation Point Flow of a Nanofluid with Velocity Slip, Non-linear Radiation and Newtonian Heating

The flow of a viscous, incompressiable and electrically conducting nanofluid flow over a stretching sheet under the influence of a transverse magnetic field is investigated taking in to account the effect of non-linear thermal radiation, newtonian heating and partial velocity slip. The nanofluid model considered in the paper incorporates the effect of Brownian motion and thermophoresis. The governing equations, in similarity form, are solved using Matlab's in-built boundary value problem solver “bvp4c”. The nanofluid flow model discussed in the present paper has significant applications in fluid engineering devices where the boundary surface is subjected to convecting heating and the temperature difference between the ambient fluid and the surface is large.

On Fire-Proof Paint Thickness for Steel under Transient Fire

Fire-proof paint thickness is of importance to steel structure fire safety, and the common programme iteratively solves its heat conduction equation in fire-insistence test, while by dimensional analysis in the equation, this paper constructs a function of steel structure section-material grouped parameter, and also a similarity form is employed to form a relationship between any two different transient fires, then solves the grouped parameters in two other transient fires. Parameter comparisons among these 3 fires conclude: with the same fire time, fire-proof paint and size, fire-proof paint thickness is necessary to be the biggest in fire-insistence test, and smaller in site fire, and the smallest in t-square fire. The key factor to decide the thickness is surface temperature outside fire-paint. The common programme is with some redundance, and more reasonable guard should add to the steel structure in t-square fire. The foregoing analysis to fire-proof paint thickness by the function of steel structure section-material grouped parameter and the similarity form between any two different transient fires, broadens the common programme to any transient fires.

Mixed convection boundary-layer flow on a horizontal flat surface with a convective boundary condition

The steady mixed convection boundary-layer flow on an upward facing horizontal surface heated convectively is considered. The problem is reduced to similarity form, a necessary requirement for which is that the outer flow and surface heat transfer coefficient are spatially dependent. The resulting similarity equations involve, apart from the Prandtl number, two dimensionless parameters, a measure of the relative strength of the outer flow M and a heat transfer coefficient γ. The free convection, M=0, case is considered with the asymptotic limits of large and small γ being derived. Results for the general, M>0, case are presented and the asymptotic limit of large M being treated.

Similarity Form, Similarity Variational Element and Similarity Solution to (2+1) Dimensional Breaking Solition Equation

In this paper, the thoughts and methods of ordinary deferential equation of dimensional non-linear partial deferential equation with (2+1) dimensional non-linear partial deferential equation will be analysed first, then the authors deduce the universal similarity form and similarity variational element of (2+1) dimensional breaking solution equation and its suitable deferential equation and find the similarity solition to (2+1) dimensional breaking solition equation.

Viscous exchange flows

Gravitationally driven exchange flows of viscous fluids with different densities are analysed theoretically and investigated experimentally within a horizontal channel. Following initiation from rest when there is a vertical boundary dividing the two fluids, the denser fluid slumps under the less dense along the underlying boundary, while the less dense fluid intrudes along the upper boundary. The motion is driven by the pressure gradients associated with the density differences between the two fluids, resisted by viscous stresses, and mathematically modelled by a similarity solution that depends on the ratio of the viscosities of the two fluids. When the viscosity of the less dense fluid is much smaller than the viscosity of the denser fluid, the shape of the interface between the fluids varies rapidly close to the upper boundary and depends weakly on the viscosity ratio within the interior of the flow. Matched asymptotic expansions are employed in this regime to determine the shape of the interface and the rates of its propagation along the boundaries. The similarity solutions are shown to be linearly stable and thus are expected to represent the intermediate asymptotics of the flow. Experiments confirm the similarity form of solutions and demonstrate close agreement with the theoretical predictions when the viscosities of the fluids are comparable, but exhibit some discrepancies when the viscosities differ more substantially. It is suggested that these discrepancies may be due to mixing between the fluids close to the boundaries, which is induced by the no-slip boundary condition. Exchange flows within porous domains are also investigated to determine the shape of the interface as a function of the ratio of the viscosities of the two fluids and using asymptotic analysis, this shape is determined when this ratio is much larger, or smaller, than unity.

Exact solution of a boundary value problem describing the uniform cylindrical or spherical piston motion

In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill [1], we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill [1]. We find that although the approximate solution of Lighthill [1] gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s [1] solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions.

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5–8] it was assumed that the convective heat transfer parameter h f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.

Radial Spreading of a Surfactant on a Thin Liquid Film

When a surfactant is placed on a layer of fluid, it reduces surface tension locally, creating a surface stress imbalance that sets the fluid in motion. The lubrication approximation is applied to axisymmetric spreading, yielding a coupled system of nonlinear partial differential equation for the height of the fluid free surface and the distribution of the surfactant. For a simplified system ignoring the effects of gravity and capillarity, as well as diffusion of surfactant molecules, the location of the surfactant can be tracked numerically. The free surface height converges quickly to a similarity form [Jensen, “Self-similar, surfactant-driven flows.” Physics of Fluids 6 (1994): 1084–94] away from the origin. Near the origin, a self-similar solution is identified, but it differs qualitatively from long-time numerical solutions. Including nonself-similar terms in an expansion around the origin corrects this inconsistency.

Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation

A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 ⩽ R a b ⩽ 1 —the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0 ⩽ θ → ∞ , the similarity solution shows that when R a b < 0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0 ⩽ θ ∼ 2 , reasonable agreement in predictions extents up to R a b ∼ 0.7 . In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.

Similarity consideration of the buoyant jet resulting from hydrogen leakage

In this paper, the similarity form is developed of a hydrogen–air buoyant jet ‘forced plume’ resulting from an unignited small-scale hydrogen leakage in the air. As the domain temperature is assumed to be constant and therefore the density of the mixture is a function of the concentration only, the hydrogen–air mixture is assumed to be of a linear mixing type. The rate of entrainment is assumed to be a function of the plume centerline velocity and the ratio of the mean plume and ambient densities. The local rate of entrainment may be considered to be consisted of two components; one is the component of entrainment due to jet momentum while the other is the component of entrainment due to buoyancy. Consequently, the entrainment coefficient is taken as a variable not constant. The density will be considered as a variable in all terms of the equations of continuity, momentum and concentration. The integral models of the mass, momentum and concentration fluxes are obtained. Non-dimensional transformations known as similarity transformations and used to transform the integral model to a set of ordinary differential equations called similarity equations which are solved numerically. Under various values of Froude number, the hydrogen centerline mass fraction, jet width and centerline velocity are introduced and compared with published experimental results for a slow leak vertical hydrogen jet.

Exact vortex solutions of the Navier–Stokes equations with axisymmetric strain and suction or injection

New solutions of the Navier–Stokes equations are presented for axisymmetric vortex flows subject to strain and to suction or injection. Those expressible in simple separable or similarity form are emphasized. These exhibit the competing roles of diffusion, advection and vortex stretching.

The dam-break problem for viscous fluids in the high-capillary-number limit

Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity μ ≈ 350 Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier–Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flow-depth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

Estudo florístico do componente herbáceo e relação com solos em áreas de caatinga do embasamento cristalino e bacia sedimentar, Petrolândia, PE, Brasil

Realizou-se um levantamento florístico em áreas de caatinga do embasamento cristalino e de bacia sedimentar, em Petrolândia, Pernambuco, objetivando-se identificar diferenças na composição, hábito e forma de vida das herbáceas entre as duas áreas. Foram coletadas amostras de solo nas profundidades de 0 a 20 cm e de 20 a 40 cm. Foram estabelecidas 100 parcelas de 1×1 m em cada área e coletadas todas as espécies herbáceas. De modo geral, o solo da área do embasamento cristalino foi mais fértil, raso, com maior capacidade de reter água e maior porcentagem de argila. Os números de famílias, gêneros e espécies das áreas da bacia sedimentar e do embasamento cristalino foram 32, 62 e 78 e 31, 53 e 69, respectivamente. A similaridade florística entre as áreas foi alta (75%). A análise de agrupamento mostrou que os arranjos florísticos entre as parcelas amostradas são mais semelhantes dentro de cada área que entre áreas, sugerindo a existência de grupos distintos. Tipo de solo parece não ser um fator decisivo na similaridade florística, hábito e proporção de forma de vida das herbáceas da vegetação da caatinga quando as áreas são próximas, mas pode influenciar a ocorrência e a freqüência de algumas espécies.

Folding Mechanism of Reduced Cytochrome c: Equilibrium and Kinetic Properties in the Presence of Carbon Monoxide

Despite close structural similarity, the ferric and ferrous forms of cytochrome c differ greatly in terms of their ligand binding properties, stability, folding, and dynamics. The reduced heme iron binds diatomic ligands such as CO only under destabilizing conditions that promote weakening or disruption of native methionine–iron linkage. This makes CO a useful conformational probe for detecting partially structured states that cannot be observed in the absence of endogenous ligands. Heme absorbance, circular dichroism, and NMR were used to characterize the denaturant-induced unfolding equilibrium of ferrocytochrome c in the presence and in the absence of CO. In addition to the native state (N), which does not bind CO, and the unfolded CO complex (U-CO), a structurally distinct CO-bound form (M-CO) accumulates to high levels (∼ 75% of the population) at intermediate guanidine HCl concentrations. Comparison of the unfolding transitions for different conformational probes reveals that M-CO is a compact state containing a native-like helical core and regions of local disorder in the segment containing the native Met80 ligand and adjacent loops. Kinetic measurements of CO binding and dissociation under native, partially denaturing, and fully unfolded conditions indicate that a state M that is structurally analogous to M-CO is populated even in the absence of CO. The binding energy of the CO ligand lowers the free energy of this high-energy state to such an extent that it accumulates even under mildly denaturing equilibrium conditions. The thermodynamic and kinetic parameters obtained in this study provide a fully self-consistent description of the linked unfolding/CO binding equilibria of reduced cytochrome c.

Mixed Convection Boundary-Layer Flow in a Porous Medium Filled with Water Close to its Maximum Density

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water close to its maximum density is considered for uniform wall temperature and outer flow. The problem can be reduced to similarity form and the resulting equations are examined in terms of a mixed convection parameter λ and a parameter δ which measures the difference between the ambient temperature and the temperature at the maximum density. Both assisting (λ > 0) and opposing flows (λ < 0) are considered. A value δ0 is found for which there are dual solutions for a range λc < λ < 0 of λ (the value of λc dependent on δ) and single solutions for all λ ≥ 0. Another value of δ1 of δ, with δ1 > δ0, is found for which there are dual solutions for a range 0 < λ < λc of positive values of λ, with solutions for all λ≤ 0. There is also a range δ0 < δ < δ1 where there are solutions only for a finite range of λ, with critical points at both positive and negative values of λ, thus putting a finite limit on the range of existence of solutions.

Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T∞)p, where T is the local temperature and T∞ is the ambient temperature, is considered when there are power-law variations in either the wall temperature or the wall heat flux which enables the equations to be reduced to similarity form. When the wall temperature is prescribed, solutions are found for p ≤ 2 and p ≥ pc (pc = 10.673) with a saddle-node bifurcation at p = pc and two solution branches for p > pc. When the wall heat flux is prescribed, solutions are found only for p < 2. The special case p = 2 is considered and the limiting forms as p → 2 and p → ∞ are obtained and compared with the solutions obtained from solving the similarity equations numerically

Two-dimensional inviscid pinch-off: An example of self-similarity of the second kind

The pinch-off of a two-dimensional region of inviscid fluid is investigated using numerical and analytical techniques. We find that pinch-off occurs when a sufficiently deformed 2D drop is released from rest. The asymptotic collapse of the pinching region is characterized by an anomalous, nonrational similarity exponent α, indicating the existence of self-similarity of the second kind. Numerical solutions of the boundary integral equations show that the height of the pinch region shrinks faster than the width, so that the singularity can be described by a slender approximation. The partial differential equations obtained from this approximation are solved and are consistent with the full boundary integral methods. Furthermore, by casting the partial differential equations into similarity form, we solve a nonlinear eigenvalue problem to obtain the value of the similarity exponent, α=0.6869±0.0003.