Normal Derivative Of Pressure Research Articles

MHD Stokes flow and heat transfer in a lid‐driven square cavity under horizontal magnetic field

This study considers the steady flow of a viscous, incompressible and electrically conducting fluid in a lid‐driven square cavity under the effect of a uniform horizontally applied magnetic field. The governing equations are obtained from the Navier‐Stokes equations including buoyancy and Lorentz force terms and the energy equation including Joule heating and viscous dissipation terms. These equations are solved iteratively in terms of velocity components, stream function, vorticity, temperature, and pressure by using radial basis function approximation. Particular solution, which is approximated by radial basis functions to satisfy both differential equation and boundary conditions, becomes the solution of the differential equation itself. Vorticity boundary conditions are obtained from stream function equation using finite difference scheme. Normal derivative of pressure is taken as zero on the boundary. The numerical results are obtained for several values of Hartmann number and Grashof number for the Stokes approximation (Re << 1). The results show that when the viscous dissipation is present, the flow and isolines concentrate through the cold wall forming boundary layers as Grashof number increases. An increase in the magnetic field intensity retards the effect of buoyancy force in the square cavity, whereas the movement of the upper lid causes buoyancy force to be dominant. The solution is obtained in a considerably low computational expense through the use of radial basis function approximations for the MHD equations. Copyright © 2017 John Wiley & Sons, Ltd.

The derivative-free Fourier shell identity for photoacoustics.

In X-ray tomography, the Fourier slice theorem provides a relationship between the Fourier components of the object being imaged and the measured projection data. The Fourier slice theorem is the basis for X-ray Fourier-based tomographic inversion techniques. A similar relationship, referred to as the ‘Fourier shell identity’ has been previously derived for photoacoustic applications. However, this identity relates the pressure wavefield data function and its normal derivative measured on an arbitrary enclosing aperture to the three-dimensional Fourier transform of the enclosed object evaluated on a sphere. Since the normal derivative of pressure is not normally measured, the applicability of the formulation is limited in this form. In this paper, alternative derivations of the Fourier shell identity in 1D, 2D polar and 3D spherical polar coordinates are presented. The presented formulations do not require the normal derivative of pressure, thereby lending the formulas directly adaptable for Fourier based absorber reconstructions.

A sloping-boundary condition for PE calculations in range-dependent acoustic media

The traditional one-way parabolic equation (PE) formulation for range-dependent layered acoustic media is modified to include effects associated with the boundary conditions along a sloping interface. Essentially, the boundary condition for continuity of the normal derivative of pressure along a sloping interface can be cast in an approximate form which does not depend on range but does contain terms up to second order in the derivatives with respect to depth. The new sloping-boundary condition is then applied along an ‘‘equivalent’’ horizontal interface within each range-independent step of the PE. Numerical results obtained for the ASA benchmark-wedge problem are used to test the new approach against an accurate two-way PE algorithm. Good agreement in this case suggests that the sloping-boundary condition, incorporated into a one-way PE, can lead to an efficient and accurate alternative to forward calculations using two-way PE.

A new integral equation formulation for an axisymmetric structure

A new method is developed to compute the acoustic field outside an axisymmetric structure from the normal velocity values on the surface. Surface pressure and normal velocity are expanded in a series of functions that are orthonormal on the surface of the structure and have a constant ratio of pressure to normal derivative of pressure at vanishing frequency. The Helmholtz integral equation is next used to compute the field everywhere outside the structure. The method is tested by applying it to scattering from a rigid cylinder with hemispherical endcaps. The series is shown to converge very rapidly.