Determination of mean pressure from PIV in compressible flows using the Reynolds-averaging approach

Abstract

The feasibility of computing the flow pressure on the basis of PIV velocity data has been demonstrated abundantly for lowspeed conditions. The added complications occurring for high-speed compressible flows have, however, so far proved to be largely inhibitive for the accurate experimental determination of instantaneous pressure. Obtaining mean pressure may remain a worthwhile and realistic goal to pursue. In a previous study, a Reynolds-averaging procedure was developed for this, under the moderate-Mach-number assumption that density fluctuations can be neglected. The present communication addresses the accuracy of this assumption, and the consistency of its implementation, by evaluating of the relevance of the different contributions resulting from the Reynolds-averaging. The methodology involves a theoretical order-of-magnitude analysis, complemented with a quantitative assessment based on a simulated and a real PIV experiment. The assessments show that it is sufficient to account for spatial variations in the mean velocity and the Reynolds-stresses and that temporal and spatial density variations (fluctuations and gradients) are of secondary importance and comparable order-of-magnitude. This result permits to simplify the calculation of mean pressure from PIV velocity data and to validate the approximation of neglecting temporal and spatial density variations without having access to reference pressure data.

Effects of Exhaust Plume and Nozzle Length on Compressible Base Flows

“…The velocity results were used as input data to calculate the pressure fields according to a Reynolds-averaging approach [26,[48][49][50], for which the Reynolds-averaged momentum equation was formulated as follows:…”

“…(1) does not contain temporal derivatives because these vanish in the averaging procedure. Terms associated with viscosity, the spatial gradient of the mean density, as well as density fluctuations only make a negligible contribution in the pressure determination, and are therefore omitted [26]. The density is eliminated from the original momentum equation with the gas law, whereas the temperature in Eq.…”

“…The validity of assuming adiabatic flow for the purpose of pressure determination has been demonstrated for compressible flows with shocks [50] as well as for a transonic axisymmetric base flow [26].…”

“…To facilitate comparison of the different cases, the boundary condition was normalized with respect to the flow case with the longest nozzle (L∕D 1.8) without a plume. For a more detailed description of the procedure, the reader is referred to the work of van Gent et al [26]. All pressure values were normalized according to C p 2p − p ref ∕p ref γM 2 ref .…”

“…In low-speed flows, the possibility of obtaining time-resolved PIV measurements allows the reconstruction of instantaneous pressure fields. In high-speed flows, this approach is, as of yet, not feasible, but the use of ensembles of uncorrelated PIV measurements still permits reconstruction of the mean pressure field [26]. This technique of deriving the pressure from the velocity data is particularly beneficial when the model geometry imposes spatial limitations for the installation of surface pressure taps or sensors, as is the case for a model equipped with a nozzle to generate an exhaust plume.…”