**Vorticity Shedding in Vortex Flowmeters – Numerical Simulation and Experimental Verification**

**Abstract**

In the present thesis, numerical and experimental investigations are conducted to study the flow around bluff bodies in a wide range of Reynolds number. In numerical investigations, two main flow types are considered, free flow and confined flow. A single bluff body of different shapes is employed in the case of free flow. In the case of confined flow, single and dual bluff bodies are considered. Five basic bluff body shapes are used, namely circular, square, triangular, inverted triangular and T-shaped cylinders. In dual bluff bodies’ investigations, only circular cylinders are considered. The experimental investigations comprise two approaches. In the first, “Hydrogen Bubble Flow Visualization System” is used to simulate the flow around bluff body and video-camera shots are utilized to photographically store visualized streamlines. Secondly, experimental Water Channel is used to investigate the applicability of vortex shedding frequency measurement in detecting volume flow rate in a confined flow.

In the case of free flow past bluff bodies, the flow is numerically simulated in the range of Reynolds number Re10^{7}.

In the Reynolds number range (Re 40), the relation between Reynolds number and vortex length for each shape is obtained. For such low Reynolds numbers, the flow is laminar steady and a separation region is visible in the wake behind the bluff body. In this region, two symmetric vortices are formed. The size of the vortex increases with increasing Reynolds number for all shapes. Hydrogen Bubble Flow Visualization photographs are compared with numerical streamline plots and are found markedly similar.

At a certain critical Reynolds number, the steady flow past the bluff body becomes unstable and the flow bifurcates to the unsteady state. The critical Reynolds number for each shape is evaluated and the effect of bluff body geometry on the value of this critical Reynolds number is identified. The critical Reynolds number for the shapes under consideration ranges between 37 and 46.

As the Reynolds number is further increased, vortices are shed in the wake of all considered body shapes. The Strouhal number for such vortex shedding phenomena is calculated and the effect of bluff body shape on the value of this number is demonstrated in the Reynolds number range (100 Re 2 x 10^{5}). Streamlines and vorticity contours for the flow around the different bluff body shapes are plotted. Visualization photographs indicate a flow behavior which is rather similar to that numerically predicted. .

On further increasing Reynolds number, the flow becomes turbulent. Using four different schemes, namely, Spalart-Allmaras, k-, SST k-, and RSM models, turbulent flow simulations are performed in the Reynolds number range (10^{4} Re 10^{7}) and values of Strouhal number are calculated. Comparisons between results using these schemes and other published data show that the k- and SST k- models are rather suitable than the RSM and Spalart-Allmaras models for the considered cases. Streamlines and vorticity contours for the flow around the different bluff body shapes at Re = 10^{6} are plotted. Periodicity of vortex shedding in the flow field is proved to exist for all shapes under consideration.

In the case of confined flow, the Strouhal number is calculated using k-ε model for domain widths (H) equal to three and four times the frontal length of the bluff body, (Blockage ratio B = 33.3% and B = 25%, respectively). The relation between Reynolds number and Strouhal number is obtained for each shape of the five bluff bodies in the Reynolds number range of (10^{3} Re 10^{8}). Streamlines and vorticity contours for the flow around different bluff bodies are plotted. The resulted vortices are shown to generally have perfect periodicity.

Considering dual circular cylinder bodies, the relation between Reynolds number and Strouhal number, for center to center distance of twice and three times the frontal length of either bluff bodies, is compared with that of the single body in the Reynolds number range of (10^{3} Re 10^{8}). Plotted streamlines and vorticity contours demonstrated a perfect periodicity of vortex shedding for the studied shapes.

Finally, experimental work, carried out in the water channel, is directed towards measuring the frequency of vortex shedding in the wake of bluff bodies with frontal length one fourth the domain’s width. Knowing the value of Strouhal number from numerical results (case of confined flow) and using the measured frequency of vortex shedding, the volume flow rate in the channel is determined and a discharge coefficient is established for every shape of bluff bodies. Results indicate that the triangular shape is the most convenient geometry for use in vortex shedding flowmeters.

**Supervisors:** Prof. Dr. Kamel Elshorbagy.

Ass. Prof. Essam Wahba.