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Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Dimensionless numbers SOE2156: Fluids Lecture 5 Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Dimensions Physical quantities have dimensions { i.e. units { associated with them. Distance can be measured in metres millimetres miles light-years but always has dimensions of Length For an equation describing a physical situation to be true, the two sides must be equal both numerically and dimensionally 1 elephant + 3 aeroplanes = 4 days but 1 metre + 3 metres = 4 metres Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers In general, if A + B + C + ::: = Z then A : : : Z must all have same dimensions { dimensionally homogeneous. (Use this to check equations!) Could have dimensions for everything { force, voltage, frequency. However SI system prescribes 6 base quantities : Length L Mass M Time T Current I Temperature Luminous Intensity Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers All other quantities can be expressed in terms of these { force F = ma, dimensions [F ] = [M ][LT 2] = [MLT 2] Dimensionless equations can be important quantities as well. Eg. an ellipse : a b gives the shape of the ellipse b a b a Similarly in uid dynamics Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Geometric similarity 2 objects are geometrically similar if all their dimensions are in the same proportion B b a A Here a b = A B { Typically this might be a real object and a scale model for testing. Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers If 2 objects are geometrically similar, same Reynolds number the ow patterns will be the same as will be the forces : : : if expressed as dimensionless numbers If an object in a ow (e.g. a car) experiences a force F from the ow, this force will depend on the xsect area A, and the uid ow speed V and density . However, the quantity C= F =A is dimensionless V2 1 2 Hence C will be the same for any geometrically similar objects at the same Reynolds number Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Other examples { for 2 geometrically similar pumps, the head and ow coecients KH = gH N 2D 2 KQ = will be the same. For a turbine, the power coecient KP = would be important P N D 5 3 Q ND 3 Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Reynolds numbers V L Choose a characteristic length L and ow speed V : L3 = a mass V2 = an `acceleration' L 2 so L3 VL = Inertial force Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Similarly, since F @u @ ux V = x ; and L A @y @y then the viscous force is so V F = L2 L Inertial force = V 2L2 Viscous force (V =L)L2 Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Similarly, since F @u @ ux V = x ; and L A @y @y then the viscous force is so V F = L2 L Inertial force = V 2L2 Viscous force (V =L)L2 = VL Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers Deriving dimensionless numbers How do we derive dimensionless numbers (sometimes called dimensionless groups)? Like this : 1 Identify the parameters that will make up the number (i.e. the parameters of importance in the problem) A D . (There will be up to 4). 2 Express the dimensions of each parameter in terms of the base dimensions M ; L; T . 3 The dimensionless number will be of the form Aa B b C c D d = dimensionless 4 We need to nd a d !! Dimensionless numbers Dimensions Geometric similarity Reynolds numbers Deriving dimensionless numbers However, for each base dimension M ; L; T the powers a d must sum to zero 6 This gives a set of simultaneous equations that can be solved NB. We end with 3 simultaneous equations for 4 unknowns { no unique solution. This gives us an equation of the form 5 AB a CD i.e. AB CD = dimensionless is the dimensionless number