Computation of Pressure Distribution Using PIV Velocity Data

Particle image velocimetry (PIV) can be used for pressure assessment. This non intrusive measurement method provide a high spatial resolution which is unavailable when using pressure transducers. The velocity vector information was used to solve inversely the Navier-Stokes equation to provide the pressure gradient needed for the Neumann boundary condition. Then the velocity data was used again to solve the pressure Poisson equation. Two flow problems were tested, water flow in a pipe with a constriction and an impinging air jet. The constriction flow was chosen to represent laminar flow problem where pressure is changing significantly. Results were compared with an inviscid solution. and the effect of spatial resolution was examined. The impinging air jet was chosen to represent a turbulent flow problem. The pressure distribution on the impingement plate was compared with reported data from the literature and some aspects of time averaging were discussed. and turbulent flows. The pressure field in both laminar and turbulent flows is useful when forces are to calculated, when comparison with pressure gauging is required, and when better understanding of the flow problem is desired. Pressure gauges used for pressure measurement are expansive, large in their size, and require an actual contact with the fluid. When the measurement is performed intrusively the flow and pressure are affected by the procedure itself (e.g. while using a catheter for coronary procedures). It was suggested, therefore, to avoid such difficulties by computing the pressure field from the velocity data generated by the non intrusive particle image velocimetry method. In the current study we present calculation results of pressure fields based on the instantaneous velocity fields obtained by particle image velocimetry. The calculation of the pressure field is determined by using the pressure Poisson equation which is derived by applying the divergence operator on the incompressible Newtonian Navier-Stockes equations, Introduction Particle Image Velocimetry (PIV) generates v 2 p = -PV {V mb in Q (1) instantaneous velocity maps in a two dimensional cross section of flow problems. The spatial resolution and the accuracy of the measurement, if performed adequately, are considered to be high. The measured velocity can then be used for a wide range of post where P is the piazometric pressure, V is the velocity vector, and p is density. The representation of the two dimensional form of Eq. 1 in Cartesian coordinates is, processing calculations, including velocity duav magnitude and direction, velocity gradient, 2 = { ( + 2 ( --) + [ z )~ } a) viscous shear, stream function, vorticity, and others. Mean and fluctuating components can ayh be calculated based on multiple realizations to represent the statistical parameters of unstable where u and v are the x and y components of V. The right hand side of Eq. 1 can be directly calculated fiom the velocity vector field generated by the PIV system. Gresho and Sani (1987) have pointed out that the physical boundary conditions are to be derived from the conservation of momentum, namely the Navier Stokes equations. The correct boundary conditions for incompressible flow are, therefore, the Neumann boundary conditions rather than the Dirchelet conditions. Similar to the right hand side of the Poisson equation (Eq. l), particle image velocimetry can be used to provide the required boundary pressure conditions by applying the Navier Stokes equations on the boundary, I? , as follows, where p is the dynamic viscosity and t is time. When either steady laminar flow or steady mean turbulent flow are to be analyzed, the time derivative, dV/dt, is zero and the PIV data can be used to compute the p{v ~VV) and pv2V terms. When turbulent flows are of interest, the Reynolds stress term is added to both Eqs. 1 and 2. The Neumann boundary conditions are obtained by solving the time average turbulent eauations and the turbulent Poisson eauation I is hence, The pressure field calculation was tested for two flow problems; water flow in a pipe with a constriction and an impinging air jet. The constriction flow was chosen to represent laminar flow problem where pressure is changing significantly. Results were compared with an inviscid solution and the effect of spatial resolution was examined. The impinging air jet was chosen to represent a turbulent flow problem. The pressure distribution on the impingement plate was compared with reported data from the literature and some aspects of time averaging were discussed. Fig. 1 The measurement regions. index notation was used such that Experimental Constriction Flow A constant distilled water i, j = 1,2,3 and ui and xi represent the flow rate of 360 rnllmin was circulated velocity and location. Here V denotes the though a 20inner diameter glass tube time averaged velocity vector. The Cartesian with a 10inner diameter constriction. A representation of V d/& (ulu;) is, 5pm Polyamid seeding particles (PSP-5, Dantec) solution was added to the distilled water, resulting in a concentration of 116 mgll. A 160mJ per pulse Nd:YAG double laser system (Quantel), a cross correlation lKxlK CCD camera (Kodak, MEGAPLUS ES 1.0), an image acquisition system (OFS), and a home made analysis software was used for the particle image velocimetry. Mass flow rate was measured using a Coriolis acceleration flow meter (Micro-Motion). Water was circulated using a centrifugal pump and a constant head container to maintain a steady flow rate. Fully developed flow was achieved using an entry length of 21 diameters. The tube orientation was vertical and flow direction was upwards to avoid accumulation of air bubbles and sedimentation. Fig. 1 shows the glass tube. The velocity vector field was generated by cross correlation within the two rectangular frames shown in Fig. 1. Calibration resulted in a micron to pixel ratio of 14.347 pdpixel. Interrogation areas of 32x32 pixels (459.1 pm x 459.1pm) and 64x64 pixels (918.2pm x 918.2pm) were used. The effect of measurement resolutionwas tested by repeating the PIV analysis every 4, 8, 16, 32, and 64 pixels (57.4, 114.8, 229.6, 459.1, and 918.2 m ) . Results showed that vector validation and filtering was unnecessary. Filters such as signal to noise ratio and a local kernel comparison resulted in zero rejection. Forty realizations of the instantaneous velocity fields were obtained and averaged for each experiment. Impinging air iet The experimental setup consists of an air supply system, an aerosol generator, a mixing chamber and a convergence section, and a round smooth glass tube (length 300 mm, inner diameter 29.5 mrn). A round flat plate (200 mm in diameter) was installed perpendicular to the flow, 1, 3, and 5 tube diameters downstream from the tube exit. A flow rate of 6.83e4 m3/s was used to meet the flow conditions of Peper et. a1 (1997) and was measured using a Micro-Motion coriolis based flow meter. Propylene-glycol particles were generated by a Laskin aerosol generator resulting in an average diameter of 0.75 mp (Echols and Young, 1963). A choice of 32x32 pixels square interrogation areas, a 57.8 mp /pixel ratio, and repeating the PIV analysis every 16 pixels resulted in 3906 vectors in a field of 57.3 mm by 58.3 mm. For each distance between the tube exit and the plate (Idi, 3di, and 5di, with di representing the nozzle diameter), 130 realizations of the instantaneous velocity fields were measured. Vector validation was obtained by a signal to noise filter and a local kernel comparison filter resulting in an average rejection rate of approximately 5%. One such velocity map is shown in Fig. 2. Fig. 2 An impinging jet velocity vector map. Fig. 3 A tube velocity vector map. -30 -25 -20 -15 -10 -5 0 I