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is the Reynolds number with the cylinder diameter as its characteristic length; Pr {\displaystyle \Pr } is the Prandtl number . The Churchill–Bernstein equation is valid for a wide range of Reynolds numbers and Prandtl numbers, as long as the product of the two is greater than or equal to 0.2, as defined above.
L = clearance length (metre) As can be seen from the formula, the clearance height c has much more influence on the leakage than the length. The formula clearly hints of pure laminar flow conditions. It is also valid for gases. Contact between the spool and the wall, the value that is generally used for practical calculations: Flow Q e = 2.5 · Qi
The hydraulic diameter, D H, is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channel length, it is defined as [1] [2] =, where
The size of the largest scales of fluid motion (sometimes called eddies) are set by the overall geometry of the flow. For instance, in an industrial smoke stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of the smallest scales is set by the Reynolds number.
Where is the dimensionless Strouhal number, is the vortex shedding frequency (Hz), is the diameter of the cylinder (m), and is the flow velocity (m/s). The Strouhal number depends on the Reynolds number R e {\displaystyle \mathrm {Re} } , [ 5 ] but a value of 0.22 is commonly used. [ 6 ]
Here l is the turbulence or eddy length scale, given below, and c μ is a k – ε model parameter whose value is typically given as 0.09; =. The turbulent length scale can be estimated as =, with L a characteristic length. For internal flows this may take the value of the inlet duct (or pipe) width (or diameter) or the hydraulic diameter.
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The area required to calculate the volumetric flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface. The vector area is a combination of the magnitude of the area through which the volume passes through, A , and a unit vector normal to the area, n ^ {\displaystyle {\hat {\mathbf {n} }}} .