Examination of the flow around a hot-wire probe using particle image velocimetry

Previous work has shown that the k ² term in the effective cooling velocity equation for inclined hot-wires can become negative under certain probe configurations and wire length-to-diameter ratios. It was hypothesised that this was due to a downwash component of velocity along the wire when prong interference effects were expected to be minimal. Direct measurements of the flow around a typical hot-wire probe using digital particle image velocimetry have shown that this downwash velocity component does exist, leading to negative values of k ² as calculated from the angle of deviation from the free stream.List of symbols d diameter of hot-wire mm - k factor in equation for effective dimensionless velocity for inclined hot-wire - l length of hot-wire mm - Q effective velocity mm/s - U free stream velocity mm/s - angle between free stream and degrees wire normal - angle through which flow is degrees deflected at working section of wire     

Acoustic Sounding of Vortex Rings in a Continuously Stratified Fluid

The experimental simulation of solitary vortex rings in a stratified fluid performed using high-frequency echo-sounding and optical visualization methods shows that on the range from turbulent to laminar regimes the vortex is a volume inhomogeneity with a sound scattering cross-section m _vU ⁵, where U is the translational velocity. The absolute value of m _v is determined by the microscale component of the vortex microstructure, which is commensurable with the sounding sonic wave length.     

Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves

Permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated. An application of the hodograph transform converts the free boundary-value problem into a boundary-value problem for the Cauchy-Riemann equations in the lower halfplane with nonlinear differential boundary conditions. This can be converted to an integro-differential equation with symbol –k ²+4|k|–4(1+), where is a bifurcation parameter. A normal-form analysis is presented which shows that the boundary-value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for || small. This normal form system has been studied thoroughly by several authors (Iooss &Kirchgässner [8],Iooss &Pérouème [10],Dias &Iooss [5]). It admits a pair of solitary-wave solutions which are reversible in the sense ofKirchgässner [11]. By applying a method introduced in [11], it is shown that this pair of reversible solitary waves persists for the boundary-value problem, and that the decay at infinity of these solitary waves is at least like 1/|x|.     

Effect of suction and injection on flow along a vertical plane surface

This study centres round the problem of flow of a liquid past a vertical porous flat plate. Considering two cases, when the plate is stationary and when it is in motion, the effect of porosity on the flow has been determined. It is found that, when the plate is stationary, the velocity of the liquid increases with increase in the suction velocity and decreases with increase in the injection velocity, and for a given suction or injection velocity, the velocity of the liquid increases with increase in time and approaches to the steady state case. But, when the plate is in motion, the velocity of the liquid decreases with increase in the suction velocity and increases with increase in the injection velocity in the constant film thickness region and also in the dynamic meniscus region provided that the gravitational force is greater than the surface tension force. In this case, the stagnation point and the minimum pressure point on the free surface have also been determined. In the case of injection there always exists a unique stagnation point and also a minimum pressure point. But in the case of suction the stagnation point does not always exist and there is no minimum pressure point.Nomenclature A _n roots of equation (3.18) - C function defined by equation (4.20) - C _n coefficients defined by equation (4.15) - F function of R ₀ and T ₀ defined by equation (4.23) - g acceleration of gravity - h film thickness at any point - h ₀ film thickness in the constant thickness region - h _m film thickness at the minimum pressure point - h _st film thickness at the stagnation point - L _m location of the minimum pressure point=h _m /h ₀ - L _st location of the stagnation point=h _st/h ₀ - n summation index - N function defined by equation (4.11) - p pressure - q flow rate - q ₀ flow rate in the constant thickness region - Q non-dimensional flow rate - R suction or injection Reynolds number=v ₀ h ₀/v - R ₀ suction or injection Reynolds number corresponding to the constant thickness region=v ₀ h/ - t time - T non-dimensional time=t/h ² - T ₀ non-dimensional parallel flow film thickness=h ₀(g/u _w )^1/2 - u vertical velocity - u perturbation velocity for u - u _s surface velocity - u _W withdrawal velocity of the plate - U steady part of the velocity u for the stationary plate - non-dimensional velocity=u/gh ² - U* non-dimensional velocity=U/gh ² - v horizontal velocity - v perturbation velocity for V - v ₀ velocity of suction or injection - V transient part of the velocity u for stationary plate - x, y coordinates - X non-dimensional x-coordinate=x ²/gh ⁴ - Y non-dimensional y-coordinate=y/h Greek Symbols _n roots of equation (3.14) - _n eigenvalues defined by equation (4.13) - _n functions defined by equation (4.14) - _n eigenvalues defined by equation (3.15) - _n non-dimensional eigenvalues= _n h/ - kinematic viscosity - liquid density - surface tension of the liquid air interface - stream function - non-dimensional stream function=/gh ³     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lie's classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q ⁽³⁾=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q ⁽³⁾=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q ⁽³⁾==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n5) ODEs which admitnontrivial contact symmetry algebras.     

The motion of viscous liquid column with finite length in a vertical straight capillary tube

This paper deals with the motion of viscous liquid column with finite length and two free surfaces in a vertical straight capillary tube. It is assumed that fluid is Newtonian. Linearizing the boundary conditions, analytic expressions in the form of infinite series have been obtained for velocity, piessure and free surface at low Reynolds number. The numerical calculation is carried out for a set of cylinder’s length of water and blood. It has been revealed that there are considerable circulating currents at the upper and lower meniscuses. Its maximum velocity is about 57% of the average velocity of the mainstream. Iner-tial effect is also studied in this paper. Using the time-dependent method in finite difference techniques, numerical solution of the corresponding nonlinear equation at Re<24.5 is computed. Comparing it with analytic exact solution at low Reynolds number shows that inertial effect is negligible provided Re<24.5.     

Solitary waves at the interface of a two-layer fluid

In this paper, we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigid walls, taking the effect of surface tension into account. First of all, we establish the basic equations suitable for the model considered, and hence derive the Korteweg-de Vries (KdV) equation satisfied by the first-order elevation of the interface with the aid of the reductive perturbation method under the approximation of weak dispersion. It is found that the KdV solitary waves may be convex upward or downward. It depends on whether the signs of the coefficients and of the KdV equation are the same or not. Then we examine in detail two critical cases, in which the nonlinear effect and the dispersion effect cannot balance under the original approximation. Applying other appropriate approximations, we obtain the modified KdV equation for the critical case of first kind (=0), and conclude that solitary waves cannot exist in the case considered as >0, but may still occur as <0, being in the form other than that of the KdV solitary wave.As for the critical case of second kind (=0), we deduce the generalized KdV equation, for which a kind of oscillatory solitary waves may occur. In addition, we discuss briefly the near-critical cases. The conclusions in this paper are in good agreement with some classical results which are extended considerably.     

Efficient simulation of free surface flows with discrete least-squares meshless method using a priori error estimator

In this article, a priori error estimate is employed to improve the efficiency of simulating free surface flows with discrete least-squares meshless (DLSM) method. DLSM is a fully least-squares approach in which both function approximation and the discretisation of the governing differential equations is carried out using a least-squares concept. The meshless shape functions are derived using the moving least-squares (MLS) method of function approximation. The discretised equations are obtained via a discrete least-squares method in which the sum of the squared residuals are minimised with respect to unknown nodal parameters. The governing equations of mass and momentum conservation are solved in a Lagrangian form using a pressure projection method. The proposed simulation strategy is composed of error estimation and a node moving refinement method. Since in free surface problems, the position of the free surface is of primary interest, a priori error estimate is used which automatically associates higher error to the nodes near the free surface. The node moving refinement method is used to construct a nodal configuration with dense nodal arrangement near the free surface. Four test problems namely dam break, evolution of a water bubble, solitary wave propagation and wave run-up on slope are investigated to test the ability and efficiency of the proposed efficient simulation method.     

Rossby solitary waves excited by the unstable topography in weak shear flow

A new forced KdV equation including topography is derived and the numerical solutions are given. The topographic variable should be related with the temporal and spatial function, which is called unstable topography. The physical features of the solitary waves about the mass and energy are discussed by theoretical analysis. In further studies, the pseudo-spectral numerical methods are used to discuss the evolution of solitary wave generated by the topography when meridional wave number \(m=1\); in a similar way, we analyze the solitary wave when meridional wave number \(m=2\). At last, we make the comparison for the characteristics of waves between \(m=1\) and \(m=2\), the wave of meridional number \(m=1\) plays a leading role.     

Determination of gradients of flow variables behind three-dimensional stationary and pseudo stationary curved shock waves in conducting gases

Summary In this paper we have obtained the gradients of magnetic field, velocity, pressure and density behind a shock wave in three dimensional steady motion of a conducting gas. For the shock configuration, we take a continuous differentiable function of coordinates and it is assumed that the components of the magnetic field H ⁱ , velocity components u ⁱ , pressure p and density behind the shock-surface are differentiable functions. Moreover we take H ⁱ , u ⁱ , p and in front of the shock-wave as constant quantities. In § 4 we have obtained the gradients of flow and field quantities behind the pseudostationary shockwave. § 5 is devoted to the calculation of gradients of flow and field quantities in cases where the normal component of the magnetic field is zero on both sides of the shock wave. In § 6 the relation between the curvature k of the shock-surface and the curvature K of the stream line just behind the shock surface in two dimensional steady motion has been derived. § 7 deals with the determination of the ratio K/k for an attached shock in the case of a wedge.     

Waves on the free surface of a two‐phase medium

A boundaryvalue problem is posed to determine the wave motion caused by propagation of a gravity wave on the free surface of a layer of a twophase medium. The problem is solved analytically in the linear approximation. The shape of the free surface, the phase velocity, and the frequency and damping factor of the wave are determined. An example of the solution of the problem is given.     

Waves excited by variable pressure on the free surface of a heavy liquid

An approximate solution of the initial and boundary value problem is constructed for a system of nonlinear integrodifferential equations describing the process of formation and evolution of axisymmetric waves excited on the unbounded free surface of an ideal liquid by pressure varying with time and in space in accordance with laws of a fairly general form. For a pressure force of limited power distributed over a fairly large area, formulas describing the free surface evolution on the semi-axist>0 (t is time) are obtained. Using the passage to the limit ast , the shape of the standing nonlinear wave excited by near-periodic pressure with a fairly wide frequency spectrum is found and its energy properties are studied.Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 97–108, May–June, 1996.     

Neck propagation as a shock wave

Neck propagation in the stretching of elastic solid filaments having a yield point was analyzed using the space one-dimensional thin filament governing equations developed previously by the authors and other researchers. Constitutive model for the filament was assumed to be expressible as engineering tensile stress(X) (tensile force) given as a function of elongational strain with the(X) curve having a yield point maxima followed by a minima and a breaking point greater than the yield point maxima. Also incorporated into the model is the hysteresis of irreversible plastic deformation. When inertia is taken into consideration, the thin filament equations were found to reduce to the nonlinear wave equation ² (X)/ ² =C ₁ ² X/ ² where is Lagrangean space coordinate, is time, andC ₁ is inertia coefficient. The above nonlinear wave equation yields a solutionX(, ) having a stepwise discontinuity inX which propagates along the axis. The zero speed limit of the step wave solution was found to describe the above neck propagation occurring in solid filaments. Furthermore, it was recognized that the nonlinear wave equation was known for many years to also govern the plastic shock wave which propagates axially within a metal rod subjected to a very strong impact on its end. The one-dimensional atmospheric shock wave also was known to be governed by the nonlinear wave equation upon making certain simplifying assumptions. The above and other evidences lead to the conclusion that neck propagation occurring in the extension of solid filament obeying the above(X) function can be formally described as a shock wave.     

Unsteady mixed convection on the stagnation-point flow adjacent to a vertical plate with a magnetic field

An analysis is performed to study the unsteady combined forced and free convection flow (mixed convection flow) of a viscous incompressible electrically conducting fluid in the vicinity of an axisymmetric stagnation point adjacent to a heated vertical surface. The unsteadiness in the flow and temperature fields is due to the free stream velocity, which varies arbitrarily with time. Both constant wall temperature and constant heat flux conditions are considered in this analysis. By using suitable transformations, the Navier–Stokes and energy equations with four independent variables (x, y, z, t) are reduced to a system of partial differential equations with two independent variables (, ). These transformations also uncouple the momentum and energy equations resulting in a primary axisymmetric flow, in an energy equation dependent on the primary flow and in a buoyancy-induced secondary flow dependent on both primary flow and energy. The resulting system of partial differential equations has been solved numerically by using both implicit finite-difference scheme and differential-difference method. An interesting result is that for a decelerating free stream velocity, flow reversal occurs in the primary flow after certain instant of time and the magnetic field delays or prevents the flow reversal. The surface heat transfer and the surface shear stress in the primary flow increase with the magnetic field, but the surface shear stress in the buoyancy-induced secondary flow decreases. Further the heat transfer increases with the Prandtl number, but the surface shear stress in the secondary flow decreases.     

Reflection, transmission and excitation of SH-surface waves by a discontinuity in mass-loading on a semi-infinite elastic medium

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.Nomenclature x ₁, x ₂, x ₃ Cartesian coordinates - , polar coordinates in x ₁, x ₃-plane - volume mass density - surface mass density of mass-loading - , Lamé constants - U scalar wave function, defined by (2.1) - c _S speed of propagation of uniform shear waves in bulk medium (c _S=(/)^1/2) - angular frequency - t time - k _S wave number of uniform shear waves (k _S=/c _S) - reduced specific acoustic impedance of mass-loading (=k _S /) - k _m wave number of SH-surface wave (k _m=k _S(1+ ²)^1/2) - _1,2,3 partial differentiation with respect to x _1,2,3 - ⁱ angle between x ₃-axis and direction of propagation of incident body wave - ⁱ wave number in horizontal direction of incident body wave ( ⁱ=k _S sin( ⁱ)) - ⁱ wave number in vertical direction of incident body wave ( ⁱ=k _S cos( ⁱ)) - C _1,2 complex amplitude of surface wave excited by a body wave - R reflection factor of surface wave, when surface wave is incident - T transmission factor of surface wave, when surface wave is incident - S particle displacement vector The research presented in this paper has been carried out with partial financial support from the Delfts Hogeschoolfonds.     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

Magnetohydrodynamic unsteady free convection past a hot vertical plate

The free convection flow of an electrically conducting liquid from an infinite plate has been studied in the presence of a uniform magnetic field. General expressions for the velocity field, induced magnetic field, skin-friction and temperature distribution have been obtained when the plate is a perfect conductor and its temperature varies with the law t ⁿ e ^at . The results have been presented through some graphs and tables with the magnetic Prandtl number unity as its value.     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Third-order systems: Periodicity conditions

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation

x+f(t,x,x^′,x^″)=0