Class Of Similarity Solutions Research Articles

Similarity solutions of thermal vortex rings: Vorticity–dynamics-based derivation

This study derives the classical similarity solutions for the thermal vortex ring, originally deduced by Scorer [J. Fluid Mech. 2, 583–594 (1957)] based on dimensional analysis from basic principles of vorticity dynamics. Derivations are performed by two methods: (1) by completing the impulse-dynamics analysis initiated by Turner [Proc. R. Soc. London, Ser. A 239, 61–75 (1957)] and (2) by a method of separation of variables. The former analysis elucidates various conditions (e.g., absence of buoyancy along the vortex-ring vertical axis), whereas the latter approach leads to the similarity solution only if one assumes that both the potential vorticity and the buoyancy evolve in a similar manner by sharing the same dependency on the vortex-ring size. It suggests a potentially richer behavior of a thermal vortex ring than predicted by similarity solutions.

The convective Stefan problem: shaping under natural convection

Abstract

Inertial gravity current in rectangular channels over a porous bottom: Asymptotic solutions

We consider a high-Reynolds-number Boussinesq gravity current propagating in a channel above a permeable horizontal boundary. The current (of reduced gravity g′) is released from a rectangular lock (of length x0 and height h0), and after an adjustment (slumping) stage is expected to enter into a similarity stage, on which we focus here, using a thin-layer shallow-water model. The classical analytical self-similar propagation solution predicts that the length of the current is given by xN(t)=Kt2∕3 (where K is a constant and t is time from release). The height h(x,t) and velocity u(x,t) display a similarity shape of the variable y=x∕xN(t),y∈[0,1] (x is the physical coordinate measured from the backwall of the lock). This solution, which is very useful in the analysis of gravity-current problems, is invalidated by the drainage effect into the porous bottom. Here we extend the classical similarity (basic) solutions by developing a perturbation (asymptotic) expansions about the basic solution. The expansion uses the small parameter λ which represents the ratio of the typical propagation time T=x0∕(g′h0)1∕2 to the drainage time tB (a given property of the porous bottom). The perturbation terms can be calculated analytically, and we present the results of the first-order correction. This provides useful insights about the influence of the porous boundary, as compared with the classical similarity behavior: xN(t) is shorter, the profile of u(y) is deflected to lower values at the nose, and h(y) is reduced mostly at the tail. The deviation from the basic similarity solution increases like λt. In addition, we show that the drainage influence is important in reducing the transition length from the inertial (inviscid) to the viscous regimes. We compared the analytical asymptotic leading-order solution with numerical finite-difference results for various values of λ, and found excellent qualitative agreement and fair quantitative agreement. We expect that higher-order terms will improve the accuracy of the new solution, but this additional extension was not performed in this work.

Evolution of supernovae-driven superbubbles with conduction and cooling

ABSTRACT We use spherically symmetric hydrodynamic simulations to study the dynamical evolution and internal structure of superbubbles (SBs) driven by clustered supernovae (SNe), focusing on the effects of thermal conduction and cooling in the interface between the hot bubble interior and cooled shell. Our simulations employ an effective diffusivity to account for turbulent mixing from non-linear instabilities that are not captured in 1D. The conductive heat flux into the shell is balanced by a combination of cooling in the interface and evaporation of shell gas into the bubble interior. This evaporation increases the density, and decreases the temperature, of the SB interior by more than an order of magnitude relative to simulations without conduction. However, most of the energy conducted into the interface is immediately lost to cooling, reducing the evaporative mass flux required to balance conduction. As a result, the evaporation rate is typically a factor of ∼3–30 lower than predicted by the classical similarity solution of (Weaver et al. 1977), which neglects cooling. Blast waves from the first ∼30 SNe remain supersonic in the SB interior because reduced evaporation from the interface lowers the mass they sweep up in the hot interior. Updating the Weaver solution to include cooling, we construct a new analytic model to predict the cooling rate, evaporation rate, and temporal evolution of SBs. The cooling rate, and hence the hot gas mass, momentum, and energy delivered by SBs, is set by the ambient interstellar mass density and the efficiency of non-linear mixing at the bubble–shell interface.

Asymptotic analysis of drug dissolution in two layers having widely differing diffusivities

Abstract This paper is concerned with a diffusion-controlled moving boundary problem in drug dissolution, in which the moving front passes from one medium to another for which the diffusivity is many orders of magnitude smaller. The classical Neumann similarity solution holds while the front is passing through the first layer, but this breaks down in the second layer. Asymptotic methods are used to understand what is happening in the second layer. Although this necessitates numerical computation, one interesting outcome is that only one calculation is required, no matter what the diffusivity is for the second layer.

More Solutions of Coupled Whitham–Broer–Kaup Equations

The similarity transformations method has been applied to get invariant solutions of the coupled system of Whitham–Broer–Kaup equations. Seven different classes of similarity solutions are derived by using Lie similarity reduction process. The solutions involve invariants which are mixing of time and space variables. Numerical simulation is performed to accomplish the physical relevance of profiles in each case. Eventually, different explicit solutions have been derived by assuming suitable values of arbitrary constants involved in the infinitesimals.

Polydisperse effects in jet spray flames

A laminar jet polydisperse spray diffusion flame is analysed mathematically for the first time using an extension of classical similarity solutions for gaseous jet flames. The analysis enables a comparison to be drawn between conditions for flame stability or flame blow-out for purely gaseous flames and for spray flames. It is found that, in contrast to the Schmidt number criteria relevant to gas flames, droplet size and initial spray polydispersity play a critical role in determining potential flame scenarios. Some qualitative agreement for lift-off height is found when comparing predictions of the theory and sparse independent experimental evidence from the literature.

On Blow-Out and Lift-Off of Laminar Jet Spray Diffusion Flames

ABSTRACTA laminar jet spray diffusion flame is analyzed mathematically for the first time using an extension of classical similarity solutions for gaseous jet flames. The analysis enables a comparison to be drawn between conditions for flame stability or flame blow-out for purely gaseous flames and for spray flames. It is found that, in contrast to the Schmidt number criteria relevant to gas flames, spray-related parameters also play a critical role in determining potential flame scenarios. Excellent qualitative agreement for lift-off height is found when comparing predictions of the theory and independent experimental evidence from the literature.

Self-similarity of heat transfer characteristics in laminar submerged and free-surface slot jet impingement

Laminar slot jet impingement of free-surface and submerged jets is investigated numerically regarding self-similar behavior in local heat transfer. Two different scaling behaviors, one for the stagnation region and one for the fully developed wall-jet region are shown, and the validity of the scaling is confirmed by fully resolved numerical simulations. Within the stagnation region, self-similarity is obtained if the relaxation process of the velocity profile is considered, which is described by the dimensionless nozzle-to-plate distance h/W·ReW as introduced in Rohlfs et al. (2014) for circular jets. Further, it is shown that the Prandtl number dependency can be well described by an analytical relationship obtained by solving the classical similarity solution of a viscous flow near stagnation point. For the wall-jet region, we first derive the scaling behavior based on the work of Watson (1964) for circular free surface jets. This analysis leads to an additional stretching of the spatial coordinate in the wall-tangential direction by the Reynolds number, e.g. x∗=x/(W·ReW).Finally, correlations for the stagnation region heat transfer are introduced for free-surface and submerged slot jets, dependent on the dimensionless nozzle-to-plate distance and constant heat flux/wall temperature, valid over a wide range of laminar flow conditions (i.e., 0.07⩽Pr≤1307,100⩽ReW⩽1000 and 4⩽h/W⩽20). Thereby, analytic Prandtl dependencies for stagnation point flows are employed and corresponding correlations for impinging slot jets are presented, valid in a very broad range (0.0005⩽Pr⩽4500).

On the self-similar propagation of gravity currents through an array of emergent vegetation-like obstacles

Self-similar propagation of gravity currents through vegetation-like obstruction arrays was elucidated. We conducted a theoretical analysis by using an approximate model for one-layer and two-layer situations. This model incorporates a balance between the driving buoyancy (i.e., pressure) force and the resisting obstruction-induced drag force that is proportional to uλ (where u —speed in the layer and λ —a constant). We focused our attention on solutions with λ ≥ 1. We considered both gravity currents in a deep ambient fluid (including both continuous-flux release currents and constant-volume currents) and lock-exchange currents and demonstrated that a variety of such flows are governed by physically acceptable similarity solutions. For gravity currents in a deep ambient fluid, our theoretical analysis revealed four distinct classes of similarity solutions. Class I solutions predict gravity currents with a triangular profile (i.e., linear current interface with a constant negative slope) and a front/nose position that is a linear function of time. The physical presence of such self-similar currents was reported in recent experimental observations for currents sustained by a continuous-flux release source. We showed that theoretical predictions of Class I solutions capture the behavior of these experimental currents well. Class II solutions predict gravity currents with a non-linear profile/interface and a constant height at the source. Though physically acceptable, we could not relate this class of solutions to presently known currents. Class III solutions correspond to constant-volume currents and predict a linear increase of velocity within the current toward the nose. We discussed this class of similarity solutions using previously reported experimental observations of such currents. Class IV solutions cover the rest of the parameter domain for all other continuous-flux release gravity currents (except those that fall under Classes I-III). Next, using a two-layer variation of the model, we developed physically acceptable similarity solutions for lock-exchange currents (released by a full-depth gate). We confirmed our theoretical results by performing comparisons with previously reported experimental data for such currents. These comparisons, while showing a good qualitative agreement, revealed a quantitative ambiguity on the value of λ for the dependency of the obstruction-induced drag force on u. We also extended our similarity solutions for the currents in a deep ambient fluid to the axisymmetric geometry. We expect our similarity solutions to provide useful insights for both “deep” and lock-exchange type of gravity currents propagating through aquatic vegetation-like obstacles and a guidance for adequate design of relevant experiments.

On the periodic injection of fluid into, and its extraction from, a confined aquifer

AbstractWe consider the periodic injection and extraction of fluid from a line well in a horizontal saturated aquifer of finite thickness as part of an aquifer thermal energy storage system. We focus on the case in which the injected fluid is dense relative to the original fluid in the aquifer and we explore the competition between the driving pressure and buoyancy force in controlling the dispersal of the injected fluid through the aquifer. We show that, after each cycle, a progressively larger fraction of the injected fluid is extracted, while the remainder of the injected fluid gradually migrates away from the well such that, after time $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}t$, the position of the leading edge of the injected fluid, $x_{nose}(t)$, scales as $ x_{nose}(t)\sim x_{nose}(\tau )\sqrt{t/\tau }$, where $\tau $ is the period of injection. If the fluid is extracted from the base of the layer, then, near the well, the thickness of the injected fluid at the end of the extraction cycle tends to a constant value, which decreases with injection rate. We also show that there is a class of self-similar exchange-flow solutions that develop when a saturated porous layer of thickness $H$ is in contact with a stratified fluid reservoir, filled to thickness $F_0H<H$ with relatively dense fluid, and with original reservoir fluid above this level. We show that these solutions coincide exactly with the far-field flow produced by the injection–extraction cycles. We successfully test the models with a series of analogue experiments of both the injection–extraction flow and the exchange flow using a Hele-Shaw cell. In the case that the fluid is injected and extracted from the top of the aquifer, the value $F_0$ tends to unity in all cases, although the convergence time depends on the rate and period of injection, the buoyancy speed and the vertical extent and the porosity of the aquifer. We use the model to explore how the concentration of reservoir fluid in the produced fluid varies as the system evolves from cycle to cycle, and we also examine the time required to transport a localised but distant contaminant to the production well through the far-field exchange flow. Finally, we consider the analogous axisymmetric injection–extraction flow problem, and show, through both numerical solution of the governing equations and experiment, that, although there is no simple class of similarity solutions, the fraction of injected fluid that is extracted progressively increases in each cycle.

Some plane symmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitation

The present study deals with the inhomogeneous plane symmetric models in scalar - tensor theory of gravitation. We used symmetry group analysis method to solve the field equations analytically. A new class of similarity solutions have been obtained by considering the inhomogeneous nature of metric potential. The physical behavior and geometrical aspects of the derived models are also discussed.

On the Lie point symmetries of the vector Maxwell equations in non-linear optics

The vector Maxwell equations of non-linear optics coupled to a single Lorentz oscillator is analyzed by using Lie symmetries. The symmetry are used to obtain classical similarity solutions of the equations to reduce to ordinary differential equations for the electric field intensity. Also, nth dimensional classification of Lie point symmetry subalgebras are given. Finally we obtain the conservation laws of symmetries.

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.

On Squeeze Film Damping in Microsystems

Classical hydrodynamic lubrication theory has been one of the most successful and widely used theories in all of engineering and applied science. This theory predicts that the force resisting the squeezing of a fluid between two parallel plates is inversely proportional to the cube of the fluid thickness. However, recent reports on liquid squeeze film damping in microsystems appear to indicate that experimentally measured damping force is proportional to the inverse of the fluid thickness to the first power—a large fundamental discrepancy from classical theory. This paper investigates potential limitations of lubrication theory in microsystems by theoretical and computational methods. The governing equations for a Newtonian incompressible fluid are solved subject to two-dimensional, parallel surface squeezing by an open-source computational fluid dynamics program called parallel hierarchic adaptive stabilized transient analysis (PHASTA), and by a classical similarity solution technique. At low convective Reynolds numbers, the damping force is determined as a function of the ratio of a reference film thickness H to a reference direction B along the film. Good agreement with classical lubrication theory is found for aspect ratios H/B as high as 1 despite the fact that lubrication theory requires that this ratio be “small.” A similarity analysis shows that when instantaneous convective Reynolds number is of order 10–100 (a range present in experiment), calculated damping deviates significantly from lubrication theory. This suggests that nonlinearity associated with high Reynolds numbers could explain the experimentally observed discrepancy in damping force. Dynamic analysis of beams undergoing small vibrations in the presence of a liquid medium further supports this finding.

Viscoelastic sink flow in a wedge for the UCM and Oldroyd-B models

The steady planar sink flow through wedges of angle π / α with α ≥ 1 / 2 of the upper convected Maxwell (UCM) and Oldroyd-B fluids is considered. The local asymptotic structure near the wedge apex is shown to comprise an outer core flow region together with thin elastic boundary layers at the wedge walls. A class of similarity solutions is described for the outer core flow in which the streamlines are straight lines giving stress and velocity singularities of O ( r − 2 ) and O ( r − 1 ) , respectively, where r ≪ 1 is the distance from the wedge apex. These solutions are matched to wall boundary layer equations which recover viscometric behaviour and are subsequently also solved using a similarity solution. The boundary layers are shown to be of thickness O ( r 2 ) , their size being independent of the wedge angle. The parametric solution of this structure is determined numerically in terms of the volume flux Q and the pressure coefficient p 0 , both of which are assumed furnished by the flow away from the wedge apex in the r = O ( 1 ) region. The solutions as described are sufficiently general to accommodate a wide variety of external flows from the far-field r = O ( 1 ) region. Recirculating regions are implicitly assumed to be absent.

On Similarity Solutions and Interface Reactions for a Vector-Valued Stefan Problem

In this paper first it is shown for several geometries that classical similarity solutions for particle growth exist if and only if the Stefan problem is well-posed in the sense of being mass conserving. The extension of the similarity solutions to multicomponent alloys, which makes the problem nonlinear, is illustrated by the application to a hypothetic alloy with realistic input values. The similarity solutions are based on the assumption of local equilibrium at the interface. In the second part, the assumption of local equilibrium is relaxed using a first-order interface reaction. The influence of the interface reaction on the movement of the interface and on the interface concentrations is evaluated using Finite Difference calculations. A Newton scheme is used to solve the nonlinear problem.

Self-similar dynamics of liquid injected into partially saturated aquifers

The injection of liquid from a central well into a partially saturated aquifer of finite thickness is described using similarity solutions. The solutions illustrate that injection leads to a growing zone around the source in which the fluid fills the whole depth of the aquifer. Beyond this zone, the current adjusts to the depth in the far field as the fluid slumps under gravity. The radial extent of the zone in which the aquifer is fully flooded depends on the ratio of the buoyancy-driven flow speed to the pressure-driven flow speed associated with the injection. New laboratory experiments, using a model porous medium, support the model predictions in the case of an initially unsaturated layer. The analysis is then developed to allow for a fully saturated aquifer, containing fluid of lower density than the injectate, and a further class of similarity solutions is developed. Again, these are shown to be consistent with new laboratory experiments. In concluding, we briefly consider how the results may be combined, to explore the self-similar dynamics of a relatively dense fluid injected into an aquifer which is partially saturated with fluid of smaller density.

Re–entrant corner flows of the upper convected Maxwell fluid

Steady planar flow of the upper convected Maxwell (UCM) fluid is described for re–entrant corners with angles 180°/α, where ½ ⩽α< 1. Local to the corner we consider a class of similarity solutions associated with the inviscid flow equations which arise from the dominance of the upper convective stress derivative in the constitutive equations. These solutions, first noted by Hinch, hold in an outer (core–flow) region and give stress singularities ofO(r−2(1−α)) (withrthe radial distance from the corner) and a stream function behaviour ofO(rnα). Herenis a parameter defining distinct solutions within this similarity class. We match such solutions to wall boundary layers, in which viscometric behaviour is retrieved. We discuss two types of boundary–layer structure. The first is a single–layer structure, previously noted by Renardy. This single layer occurs forn= 3 −αand has the viscoelastic balance of the constitutive equations holding uniformly within it. Here we complete previous analysis by considering the downstream case. The second type of boundary layer considered is a double–layer structure, which we discuss for the range 1 <n< 3 −α. Now the elastic balance of the constitutive equations holds within its main region, with a thinner region closer to the wall in which the relaxation terms are recovered. This structure extends the range of the core similarity solutions and has not been previously noted.

General solution of the aerosol dynamic equation: growth and diffusion processes

The dispersion of aerosol particles in a fluid media is studied considering the main mechanism for condensation and diffusion. This has been done when the technique of Lie is used for solving the aerosol dynamic equation. This method is very useful in sense that it reduces the partial differential equation to some ordinary differential equations. So, different classes of similarity solutions have been obtained. The quantity of well-defined physical interest is the mean particle volume has been calculated.