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CIVL 5076 Coastal Engineering ` Class notes CIVL 5076 Coastal Engineering By Ahmad Sana, Ph.D., P.E. Department of Civil and Architectural Engineering Sultan Qaboos University Sultanate of Oman Email: [email protected] May 1, 2014 1 1 Introduction During the World War II the field of coastal engineering evolved as a result of information required about the wave properties on certain coastal fronts. Nowadays, the knowledge gained through this field applies to the following civil engineering activities: • Ports, harbors, breakwaters, seawalls, revetments • Navigation channels • Coastal intakes and out-falls • Ocean pipelines, cables, piles • Sediment control: groins, sand bypassing, dredging, beach nourishment • Power generation by waves and tides In order to determine the wave properties and corresponding hydrodynamic processes, the following approaches are used: • Hydrodynamics: Wave theories • Empirical formulae based on experimental results • Physical Modeling • Numerical Modeling In the following chapters, one of the wave theories, i.e. small amplitude wave theory is discussed in more detail because it is one of the most popular and applicable theories for field applications. 2 2 Small Amplitude Wave Theory This wave theory was developed by Airy and due to that reason it is called as Airy wave theory as well. 2.1 Assumptions The basic assumptions for this wave theory are as follows: 1. Water is homogeneous and incompressible, wave lengths are greater than 3m so that capillary effects may be ignored 2. Flow is irrotational, i.e. no shear stress present anywhere. Thus the velocity potential φ must satisfy the Laplace equation: ∂ 2φ ∂ 2φ + = 0 (2.1) ∂x 2 ∂z 2 3. The bottom is not moving and is impermeable and horizontal, i.e. no energy transfer through the bed 4. The pressure along the air-sea interface is constant, i.e. no effect of weather related pressure differences 5. The wave amplitude is small compared to the wave length and water depth. The definition sketch shown in Fig.2.1 depicts the various wave properties of practical relevance. L 5 SWL 0 η H ζ w -5 u d ε -10 -15 z=- d -20 Figure 2.1. Definition sketch for wave parameters 3 Here, H is wave height, L; wave length, d; depth of water, u; horizontal velocity, w; vertical velocity, ζ, ε; water particle displacement with respect to the mean position, in x- and z- directions, respectively. The time period of the wave is denoted by T. 2.2 Basic Equations The water surface profile is given as: η = (H / 2)cos(kx −σt) Where, k is wave number ( = 2π / L ) and σ is angular frequency ( = 2π /T ) Since the velocity potential φ should be cyclic with horizontal position and time, and it should vary with depth, we can assume that: φ = f (z)sin(kx −σt) (2.2) Substituting Eq.(2.2) into (2.1), one can get: ∂ 2 f − k 2 f = 0 ∂z 2 The general solution to this partial differential equation is: f (z) = Aexp(kz) + B exp(−kz) Where A an B are arbitrary constants. Putting this value of the function f(z) in Eq.(2), we get: φ = {Aexp(kz) + B exp(−kz)}sin(kx −σt) (2.3) Two boundary conditions are required to find A and B. (a) The vertical velocity w at the bottom must be zero (Assumption 3): ∂φ w = = 0 ∂z z=−d From Eq. (2.3) ∂φ = k{Aexp(kz) − B exp(−kz)}sin(kx −σt) ∂z At z=-d ∂φ = k{Aexp(−kd) − B exp(kd)}sin(kx −σt) = 0 ∂z z=−d Since k and sin(kx −σt) can not be equal to zero (if they are zero, there is no wave!), we shall have always: Aexp(−kd) − B exp(kd) = 0 exp(kd) Or A = B exp(−kd) Substituting this value of A in Eq.(3) we get: φ = B exp(kd)[exp{k(d + z)}+ exp{− k(d + z)}]sin(kx −σt) Since, [exp{k(d + z)}+ exp{− k(d + z)}] = 2cosh k(d + z), we get: φ = 2B exp(kd)cosh k(d + z)sin(kx −σt) (2.4) (b) Second boundary condition (on the surface) may be derived from Bernoulli’s equation for time-varying flow in two dimensions, i.e. 4 ∂φ p 1 + gz + + (u 2 + w2 ) = 0 ∂t ρ 2 On the surface atmospheric pressure is there, i.e. p / ρ = 0. Assuming particle velocities to be small, u 2 + w2 = 0 . At the surface, z = η . Hence, ∂φ / ∂t + gη = 0, at the surface. 1 ∂φ Or η = − g ∂t z=η Considering η to be very small, we can say: 1 ∂φ η = − g ∂t z=0 At t=0, x=0 and z=0, η=H/2 gH ∂φ Or = − 2 ∂t z=0 From Eq.(2.4) ∂φ = − σ + −σ 2 B exp(kd)cosh k(d z) z=0 cos(kx t) ∂t z=0 At t=0, x=0 (crest of the wave), cos(kx −σt) = 1 So, Hg 2B exp(kd) = 2σ cosh kd Eq.(2.4) becomes: H g cosh k(d + z) φ = sin(kx −σt) (2.5) 2 σ cosh kd The velocity potential given by Eq.(2.5) can be used to find the velocities in x and z directions as will be shown later. Another useful relationship may be derived by considering the vertical component of velocity of a particle on the water surface, w, which is given as: ∂η 1 ∂φ w = but at the water surface, η = − so, ∂t g ∂t z=0 1 ∂ 2φ w = − ∂ 2 g t z=0 5 ∂φ Also, w = , we can write: ∂z ∂ 2φ ∂φ + g = 0 ∂t 2 ∂z Inserting φ from Eq.(2.5) and solving, we get the so-called dispersion relation: σ 2 = gk tanh(kd) (2.6) Since, the wave celerity C, is defined as: C=L/T=σ/k From Eq.(2.6) we get: gL 2πd C = tanh (2.7) 2π L Or gT 2πd C = tanh (2.8) 2π L And gT 2 2πd L = tanh (2.9) 2π L Eq.(2.9) is an implicit relationship by virtue of the wave length L and therefore, it can be solved by trials only. Fenton (1990) has given an approximate explicit relationship from which the wave length can be directly calculated from the available knowledge of wave period and depth of water as follows: 3 / 4 2 / 3 gT 2 σ 2 d L = tanh (2.10) π 2 g 2.3 Wave Classification by Relative Depth When the relative depth, d/L, is greater that 0.5, tanh(2πd / L) ≅ 1 and Eq.(2.7), (2.8) and (2.9) transform to the following, respectively: gL C = 0 (2.11) 0 2π gT C = (2.12) 0 2π And gT 2 L = (2.13) 0 2π This condition is called deep water condition and denoted by the subscript zero. In this condition it may be noted that the wave celerity is independent of the water depth. Another extreme condition may be found when the relative depth is less than 0.05. In this condition, tanh(2πd / L) ≅ 2πd / L , so from Eq.(2.7) or (2.8): C = gd . And from Eq.(2.9), L = ( gd )T . This condition is denoted as shallow water condition. It is obvious that in this situation the wave celerity depends only on the water depth. 6 Some interesting features of both the extreme conditions by virtue of relative depth can be observed from the graphical representation of the above relationships after some rearrangement. For example, dividing Eq.(2.8) by Eq.(2.12), we get: C 2πd = tanh (2.14) C0 L Or L 2πd = tanh (2.15) L0 L Multiplying both sides by d and rearranging, we get: d d 2πd = tanh (2.16) L0 L L Figure 2.2 shows Eq. (2.14) or (2.15) and (2.16) along-with some other relationships (which will be derived later). 0.01 0.1 1 10 10 2πd/L H/H0 1 1 n L/L0 0.1 0.1 d/L0 shallow Intermediate deep 0.01 0.01 0.01 0.1 1 d/L Figure 2.2. Various wave parameters against relative depth as per small amplitude wave theory. 7 Example 2.1 A wave tank is 193m long, 4.57m wide and 6.1m deep. The tank is filled to a depth of 5m with fresh water and a 1-m high, 4-sec period wave is generated. (a) Calculate the wave celerity and length using small amplitude wave theory. (b) Calculate the corresponding deep water wave length and celerity. Solution: Given: H = 1m T = 4 sec d = 5m (a) Using Eq.(2.9) gT 2 2πd 9.81× 42 2π × 5 L = tanh = tanh 2π L 2π L By trials we get, L = 22.2m L 22.2 C = = = 5.55m/sec T 4 (b) gT 2 9.81× 42 L 25 L = = = 25.0 m, and C = 0 = = 6.25 m/sec 0 2π 2π 0 T 4 For part (a) it may be noted that the explicit relationship for the wave length (Eq.2.10) may also be used. Using that we get: 3 / 4 2 / 3 3 / 4 2 / 3 gT 2 σ 2 d 9.81× 42 4π 2 × 5 L = tanh = tanh = 22.06 m π π 2 × 2 g 2 4 9.81 A value that differs by only 0.6% from the result obtained from Eq. (2.9). 2.4 Wave Kinematics and Pressure The horizontal and vertical components of water particle velocity u and w may be determined as u = ∂φ / ∂x and w = ∂φ / ∂z , respectively.
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