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In this paper the static pressure field of an annular swirling jet is measured indirectly using stereo-PIV measurements. The pressure field is obtained from numerically solving the Poisson equation, taken into account the axisymmetry of the flow At the boundaries no assumptions are made and the exact boundary conditions are applied. Since all source terms can be measured using stereo-PIV and the boundary conditions are exact, no assumptions other than axisymmetry had to be made in the calculation of the pressure field. The advantage of this method of indirect pressure measurement is its high spatial resolution compared to the traditional pitot probes. Moreover this method is non-intrusive while the insertion of a pitot tube disturbs the flow It is shown that the annular swirling flow can be divided into three regimes: a low, an intermediate and a high swirling regime. The pressure field of the low swirling regime is the superposition of the pressure field of the non-swirling jet and a swirl induced pressure field due to the centrifugal forces of the rotating jet. As the swirl increases, the swirl induced pressure field becomes dominant and for the intermediate and high swirling regimes, the simple radial equilibrium equation holds.

The most widely used direct method to measure the static pressure in a flow field is by insertion of a Pitot tube. However this method has some disadvantages. It disturbs the flow field and the probe dimensions are in the order of a few mm. In order to obtain sufficient spatial resolution, the flow field dimensions have to be very large compared to the probe dimensions. Since pressure and velocity are linked by the Navier-Stokes equations, the static pressure can also be measured indirectly by measuring the velocity field. The Navier-Stokes equations form a set of non-linear partial differential equations, consisting of a continuity equation, three momentum equations and an energy equation. The momentum equations are the link between the static pressure in the flow field and the velocity field. There exist 2 strategies to measure the pressure field indirectly. The first strategy is direct spatial integration of the momentum equations [

In this paper a study is made of the static pressure field of a swirling annular jet. Three components of velocity in a plane through the central axis are measured using stereoscopic PIV measurements. The pressure field is computed by solving the Poisson equation using exact boundary conditions based on the momentum equations. The mean flow field is found to be axisymmetric and hence all the source terms in the Poisson equation could be determined. This approach differs from previous studies found in literature which could not measure all source terms and additional assumptions had to be made for the unknown values or for the boundary conditions. Four different swirl cases were investigated: a non-swirling, a low, an intermediate and a high swirling jet. For a low swirling jet, the static pressure field is the superposition of the pressure field of a non-swirling jet and a swirl induced pressure field. This swirl induced pressure field originates as a balance of the centrifugal forces due to the rotating jet. As the swirl is increased, it's induced pressure field dominates and the radial momentum equation can be simplified to the simple radial equilibrium equation, which is a balance between pressure gradients in the radial direction and centrifugal forces.

A schematic view of the experimental setup is shown in _{o}_{i}_{o}_{o}_{0}_{o}_{i}^{2}/s. The dimensionless swirl number

The flow field is measured using the stereoscopic particle image velocimetry (PIV) technique. A photo of the experimental configuration is shown in

When calculating time averaged properties of a flow, an error is introduced by computing statistical quantities from a finite number of data samples. In this study

The Navier-Stokes equations who describe the motion of an incompressible, newtonian fluid are given by the continuity equation,

The above momentum equations in the axial and radial direction describe a relation between the pressure field and the velocity field. By taking the divergence of ^{2}

The assumption of axisymmetry is validated as the measurements confirm the velocity profiles are symmetric within measurement accuracy for all swirl cases. _{o}

The velocity is measured in _{p}_{i,j} in an internal point (point not on a boundary of the measurement domain) with coordinate _{i,j} are calculated as
^{2}p^{2}

The boundary condition for boundary 1 gives the equation _{i,j} = 0 (Dirichlet condition). On boundary two,

For the central axis on boundary 3 the partial derivative of the pressure is discretised using a second order upwind scheme giving for a symmetry boundary condition

Finally, the LHS of

Using a grid study in combination with Richardson extrapolation, the discretisation error is estimated to be around 5%. The discretisation of _{p}_{p}_{p}_{p}

The flow field of the non-swirling jet is shown in _{o}

The pressure field of the swirling jet with swirl number ^{2}

To study the influence of swirl on the static pressure field, let us now consider a decomposition of the pressure field into a swirl induced component _{S}_{0} as _{S}_{0}. Filling this decomposition into _{S}

Both pressure fields are shown in _{0} is dominant. Further downstream, _{o}_{S}_{0} is negligible. Comparing _{0} in

As the swirl is further increased, the static pressure in the low pressure region decreases as the azimuthal velocities increase (_{S}_{0} is much smaller. As a result,

This equation is called the simplified radial equilibrium equation as it is the balance between pressure gradients and centrifugal forces [

As the swirl is further increased, the centrifugal forces increase the radial expansion of the jet. The vortex breakdown bubble grows as the region of recirculation along the central axis increases. This yields a decrease of the azimuthal velocities near the central axis, which in turn decreases the pressure gradients in this region. Hence the sub-pressure near the nozzle decreases as can be seen by comparing _{0} and _{S}

In this paper the static pressure field of an annular swirling jet is measured indirectly using velocity field measurements. The pressure is computed by numerically solving the Poisson equation with the appropriate boundary conditions. The source terms of the Poisson equation are measured using stereoscopic-PIV. The advantages of this indirect method compared to the direct measurement by pitot tubes are an increased spatial resolution and the non-intrusive character of the technique. Four different swirl cases were investigated: a non-swirling, a low, an intermediate and a high swirling jet. For a low swirling jet, the static pressure field is the superposition of the pressure field of a non-swirling jet and a swirl induced pressure field. This swirl induced pressure field originates from the creation of radial pressure gradients to balance the centrifugal forces of the rotating jet. As the swirl is increased, the swirl induced pressure field dominates and the radial momentum equation can be simplified to the simple radial equilibrium equation, which expresses the balance between pressure gradients in the radial direction and centrifugal forces.

The authors gratefully acknowledge the funding of this work by the Onderzoeksfonds K.U.Leuven/ Research fund K.U.Leuven.

Schematic view of the experimental configuration, the measurement domain (dashed line) and the boundary conditions for solving the Poisson equation for pressure.

Photo of the experimental configuration.

Flowfield of the annular jet at

Normalised static pressure profile along the central axis. ■: data from Ko et al. [

Normalised static pressure

Decomposition of the pressure field of the annular jet at

Normalised static pressure

Decomposition of the pressure field of the annular jet at

Normalised static pressure

Decomposition of the pressure field of the annular jet at