J. Szantr Lectre No. 3 Potential flows 1 If the flid flow is irrotational, i.e. everwhere or almost everwhere in the field of flow there is rot 0 it means that there eists a scalar fnction ϕ,, z), sch that gradϕ. Sch a flow is called the potential flow, and fnction φ is called the velocit potential. ϕ ϕ We have: z ϕ z In case of potential flow of an incompressible flid the mass conservation eqation is transformed into the Laplace eqation: ρ t div ϕ ϕ ϕ z ρ ) divgradϕ ϕ 0 0 The Laplace eqation is linear, that means that the sm of its soltions is also a soltion. In practise, ver complicated potential fnctions, describing comple flows, ma be composed of the simple fnctions describing so called elementar flows.
The potential flows are particlarl sited for mathematical modelling of the flid motion in the regions otside the bondar laers and wakes, where the inflence of the flid viscosit is negligibl small. The method of formation of the complicated potential flows will be demonstrated sing the eample of twodimensional flows. In this case we have: where: ϕ, ) ϕ, ) ψ ψ, ), ) ϕ ψ - the velocit potential - the eqipotential lines - the stream fnction - the stream lines ϕ ψ
The elementar potential flows 1. The niform flow b ϕ a ϕ ) b a, ϕ The eqipotential lines: The velocit potential: b a The stream fnction: The stream lines: a b ) b a, ψ
The constant vales of the potential φ correspond to the constant vales of the radis r, i.e. the eqipotential lines are circles having a common centre.. The sorce positive or negative) ±πr r r r The sorce is a singlar point in the field of flow, in which an otflow of flid of a certain volmetric intensit takes place. This otflow is niform in all directions. In case of a negative sorce or a sink), the flid flows into the sink and disappears. Ths we have: ± r πr or: where: - the radial velocit ϕ r ± ϕ ± πr π ln r
In the artesian sstem of co-ordinates we have: r cosθ π and frther we have: ) the complete differential of the potential: dϕ where: θ r sinθ arctg π d d π ) ) π ) the complete differential of the stream fnction: dψ d d π ) π ) and frther, after integration we have: ϕ 1 π ln ) ψ π arctg Stream lines are straight lines passing throgh the sorce
Eample: Sperposition of an niform flow and a sorce The potential and stream fnction of the resltant flow are the sms of potentials and stream fnctions We assme that the niform flow is parallel to the ais. The potential: The stream fnction: ϕ ln 4π ψ arctg π ) The zero-th stream line: ψ 0 arctg 0
Soltion for the zero-th stream line: ctg π If we replace the zero-th stream line with a solid wall, the pictre of flow will not change and we obtain the flow arond a solid halfbod. The velocit components: π π ) ) The location of the stagnation point on the ais: 0 π The pressre p in an arbitrar point of the flow ma be calclated according to the Bernolli eqation: p ρ 1 1 ρ p p p p ρ
After sbsittion we get: p 1 π 4 ) π In the stagnation point we get: p π π π 1 Bt on the zero-th stream line at the point 0 we have: ± π or: 4 184 π 1,
3. The doblet the dipole) The dipole reslts from sperposition of a positive and negative sorce of the same absolte intensit. The measre of the dipole intensit is the so called moment Ma. ontrar to a sorce, a dipole has directional characteristics, becase it emits flid in a given direction and scks in flid from the opposite direction. Hence its location in space is important. For a dipole at 0, 0, directed in the positive direction, we have: The dipole potential: ϕ M π The dipole stream fnction: ψ M π
4 The stream lines for a dipole: The stream lines are circles of radii /, the centres of which lie on the ais in the points /. 4 The eqipotential lines for a dipole: The eqipotential lines are circles of radii /, the centres of which lie on the ais in the points /.
Eample: Irrotational flow arond a circlar clinder Sperposition: the niform flow the dipole
In the sstem Orθ we have: M πa where: a clinder radis a a 1 r r 4 cos θ where: θ arctg ρ p p 1 for ra i.e. on the clinder srface we have: p θ ρ p 1 4sin θ ) The components of the force acting on the clinder ma be calclated: P P a a π pθ cosθdθ 0 0 π pθ sinθdθ 0 0 This is so called d Alembert parado
The d Alembert parado means, that in a potential flow the forces eerted b the flowing flid on the immersed bodies are eqal zero, what does not agree with a common eperience. This is a direct conseqence of the smmetr of the calclated pressre field, which is smmetrical both with respect to the ais and ais. In realit, the pressre field on the clinder is asmmetrical with respect to the ais, what is shown in the diagram on the previos slide and in the sketch and photograph below, showing the calclated and eperimentall visalised flows. The differences are first of all visible on the trailing side of the clinder.