Wave (and current) forces WAVE FORCES NON-BREAKING BREAKING BROKEN WALL SAINFLOU MINIKIN GODA a.o. pm = w db/2 PILE MORISON MORISON +Slamming MORISON Circular CAISSON MC CAMUS & FUCHS MINIKIN - RUBBLE STRUCTURE VAN DER MEER VAN DER MEER VAN DER MEER FLOATING STRUCTURE Potential theory Slamming Potential theory WAVE FORCES NON-BREAKING BREAKING BROKEN WALL SAINFLOU MINIKIN GODA a.o. pm = w db/2 PILE MORISON MORISON +Slamming MORISON Circular CAISSON MC CAMUS & FUCHS MINIKIN - RUBBLE STRUCTURE VAN DER MEER VAN DER MEER VAN DER MEER FLOATING STRUCTURE Potential theory Slamming Potential theory Wave forces an a vertical wall (Sainflou) Ah 2H wave crest wave trough H H z Figure 2. Wave induced pressure on a vertical wall SWL SWL h cosh( ( )) cosh( ( ))( , , ) cos( ) cos( )cosh( ) cosh( )cosh( ( ))(cos( ) cos( ))cosh( )cosh( ( ))cos( ) cos( ) sin( ) sin( ) cos( ) cos( ) sin(cosh( )i ra g a g k z h k z hx z t t kx t kxkh kha g k z ht kx t kxkha g k z ht kx t kx t kx tkhe ee ee eee e e ee+ +u = + ++= + ++= + + ( )) sin( )2 cosh( ( ))cos( ) cos( ).cosh( )kxa g k z ht kxkh ee +=2 cosh( ( ))cos( ) cos( )cosh( )cosh( ( ))2 sin( ) cos( ).cosh( )d a g k z hp t kxt t khk z ha g t kxkh ee e| | cu c += = |c c \ .+=cosh( ( ))2 sin( ).cosh( )k z hp gz a g tkh e+= +ar = ai = a Total reflection - Linear wave theory Total reflection - Linear wave theory clapotis height from nonlinear theories 2cothHh khLtA =Ah Ah d = h Ah w = g Approx Pressure Distributions 2H h wave crest wave trough Ah = 0.66 m H H Fb Ft Fc z Figure 2. Wave induced pressure on a vertical wall gh gh SWL 1cosh( )gHkhFb = force per unit width from the hydrostatic pressure against the leeward wall. Fc = force per unit width for the wave crest in front of the wall Ft = force per unit width for the wave trough in front of the wall Stages in the development of flow (from left to right) past a circular cylinder from the rest. The speed of the stream has been increased rapidly and then kept constant Non-separated vs separated flow Drag and lift on a circular cylinder, length l Bernoullis equation 21constant along a streamline2p gz U + + =221 ; the steady drag force21 ; the lift force magnitude2D D pL L pF C A UF C A U==( )0.3DC =Low Reynolds number Critical Reynolds number _ + _ + _ + _ + _ + _ + _ + _ + _ + _ + _ + _ + beff beff ( )1.0DC ~Drag and lift on en circular cylinder U0 F FD FL Vortex shedding period To is given by the Strouhals number: St=D/(U0T0) ~ 0.2 D Flow patterns for flow over a cylinder: (A) Reynolds number = 0.2; (B) 12; (C) 120; (D) 30,000; (E) 500,000. Alternating vortex shedding make forces timedependent with typical period given by the shedding period T0 for transversal forces (lift) and T0/2 inline (drag). This adds up to the steady inline force we know from before as the drag force. (A) (B) (C) (D) (E) DRAG COEFFISIENT (2 D) Steady flow 2;12DpAFCbDU ==Strouhal number versus Reynolds number for a smooth non-vibrating cylinder 0Rev= U D00001=t f DSUfTWAVE FORCES NON-BREAKING BREAKING BROKEN WALL SAINFLOU MINIKIN GODA a.o. pm = w db/2 PILE MORISON MORISON +Slamming MORISON Circular CAISSON MC CAMUS & FUCHS MINIKIN - RUBBLE STRUCTURE VAN DER MEER VAN DER MEER VAN DER MEER FLOATING STRUCTURE Potential theory Slamming Potential theory Small diameter cylinder D < 0.2 Forces on structures with non-separated flows. The flow turns back before the vortex is generated. Potential theory Keulegan Carpenter number K = UmT/D