Steady compressible swirl flows with closed streamlines at high Reynolds numbers

Abstract

In steady axisymmetric flows in a closed swirl chamber one can distinguish between the swirl flow proper, with components normal to the meridian plane, and a secondary flow whose components lie in the meridian plane. One can trace the motion of a particle within the meridian plane. The closed path so obtained will be called a streamline, to be parametrized by a stream function ϕ. One can distinguish between the flow in a boundary layer, where the velocity gradient is large, and a core flow, where the velocity and temperature gradients are relatively small. The present article is concerned with only the core flow. In high Reynolds number flows in which the streamlines are not closed there are three quantities which are constant along the streamlines: the total enthalpy (the right-hand side of Bernoulli's equation), the entropy and the moment of momentum of the particles with respect to the axis of symmetry. These are determined by conditions in the entrance cross-section. In flows with closed streamlines these quantities are ultimately determined by the cumulative effects of viscosity and heat conductivity. Conditions expressing cumulative effects enter the analysis as integrabiliby conditions necessary for the existence of a second approximation in a development of the flow field with respect to the reciprocal of the Reynolds number. They are re-expressed as the balance equations for energy, entropy and angular momentum which are to be satisfied on all surfaces ϕ = constant. One thus obtains an algorithm which leads from expressions for the total enthalpy, the entropy and the angular momentum as functions of ϕ to the residuals in the balance equations, also computed as functions of ϕ. The functions with which this algorithm starts must be chosen such that the residuals in the balance equations become zero. The secondary flow can be arbitrarily slow only if the Prandtl number is ½. At the centre of the secondary motion the balance equations are linearly dependent. This fact introduces an additional free parameter which allows one to compute secondary flows with different speeds. The linearized algorithm has the character of a Fredholm integral equation. This suggests an iterative solution similar to a Neumann series. The particles experience periodic changes of state which can be discussed as thermodynamic cycles. Such an analysis shows that the heat inputs occur on average at lower temperatures than the heat outputs. Besides the work that maintains the swirl motion, which is provided by shear-force components normal to the meridian plane, one therefore needs additional work, provided by the shear-force components within the meridian plane, to maintain a secondary motion. Responsible for this state of affairs is the fact that particles which do not quite maintain adiabatic temperatures move within a field with a large pressure gradient caused by the swirl component. This makes the flow sensitive to disturbances in the energy balance.