5.10 Exercise 14: Island Wakes
113
5.10.2 The Reynolds Number
Flow around an obstacle such as an island becomes dynamically unstable under
certain circumstances and breaks up into a irregular turbulent wake. The transition
of laminar fl w into turbulence can be described by means of the ratio between
the nonlinear terms and diffusion of momentum. This ratio is called the Reynolds
number (Reynolds, 1883) and can be define by:
Re =
U L
A h
(5.30)
where U is the speed of the incident fl w, L is the diameter of the obstacle, and A h
is ambient horizontal eddy viscosity.
A variety of fl w regimes can develop in dependence on the magnitude of the
Reynolds number. For Re ≈ 1 the fl w is typically laminar and smoothly surrounds
the obstacle. A stationary vortex pair with central return f ow develops for Re ≈ 10.
Larger values of Re ≈ 100 leads to the formation of a turbulent wake in the lee
of the obstacle. Re >> 100 triggers a turbulent wake of organised vortices called
von K´ arm´ an vortex shedding in appreciation of pioneering work by Theodore von
K´ arm´ an (1911).
Vortex shedding occurs at a certain frequency f . The dimensioness number:
St =
f L
U
(5.31)
is known as the Strouhal number and is named after the Czech physicist Vincenc
Strouhal (1850–1922), who firs investigated the steady humming (or singing)
of telegraph wiring. There exist relationships (not replicated here) between the
Strouhal number and the Reynolds number that can be experimentally derived. It
should be noted that this Strouhal instability is believed to be the reason for collapse
of the Tacoma Narrows Bridge, Washington, on November 7, 1940.
5.10.3 Inclusion of Lateral Friction and Momentum Diffusion
Lateral friction and diffusion of momentum is required in the momentum equations
in order to simulate the development of turbulent wakes in the lee of an obstacle.
Under the assumption of uniform values of lateral eddy viscosity A h , the depthaveraged version of the lateral momentum diffusion can be formulated as:
div h (u) =
A h
h
∂
∂x
h
∂u
∂x
+
∂
∂y
h
∂u
∂y
(5.32)
div h (v) =
A h
h
∂
∂ x
h
∂v
∂x
+
∂
∂y
h
∂v
∂y
(5.33)
113
5.10.2 The Reynolds Number
Flow around an obstacle such as an island becomes dynamically unstable under
certain circumstances and breaks up into a irregular turbulent wake. The transition
of laminar fl w into turbulence can be described by means of the ratio between
the nonlinear terms and diffusion of momentum. This ratio is called the Reynolds
number (Reynolds, 1883) and can be define by:
Re =
U L
A h
(5.30)
where U is the speed of the incident fl w, L is the diameter of the obstacle, and A h
is ambient horizontal eddy viscosity.
A variety of fl w regimes can develop in dependence on the magnitude of the
Reynolds number. For Re ≈ 1 the fl w is typically laminar and smoothly surrounds
the obstacle. A stationary vortex pair with central return f ow develops for Re ≈ 10.
Larger values of Re ≈ 100 leads to the formation of a turbulent wake in the lee
of the obstacle. Re >> 100 triggers a turbulent wake of organised vortices called
von K´ arm´ an vortex shedding in appreciation of pioneering work by Theodore von
K´ arm´ an (1911).
Vortex shedding occurs at a certain frequency f . The dimensioness number:
St =
f L
U
(5.31)
is known as the Strouhal number and is named after the Czech physicist Vincenc
Strouhal (1850–1922), who firs investigated the steady humming (or singing)
of telegraph wiring. There exist relationships (not replicated here) between the
Strouhal number and the Reynolds number that can be experimentally derived. It
should be noted that this Strouhal instability is believed to be the reason for collapse
of the Tacoma Narrows Bridge, Washington, on November 7, 1940.
5.10.3 Inclusion of Lateral Friction and Momentum Diffusion
Lateral friction and diffusion of momentum is required in the momentum equations
in order to simulate the development of turbulent wakes in the lee of an obstacle.
Under the assumption of uniform values of lateral eddy viscosity A h , the depthaveraged version of the lateral momentum diffusion can be formulated as:
div h (u) =
A h
h
∂
∂x
h
∂u
∂x
+
∂
∂y
h
∂u
∂y
(5.32)
div h (v) =
A h
h
∂
∂ x
h
∂v
∂x
+
∂
∂y
h
∂v
∂y
(5.33)