42
J. Grue
Ekman [2]. Our purpose is to calculate as exactly as possible the internal wave (dead
water) resistance, the wave wake and speed of the FRAM, for the conditions in the
field. The computations illustrate and enhance the accuracy of the observations.
We first derive from the empirical descriptions of Nansen [1] and Ekman [2] the
dead water resistance on FRAM. Then we compute by a strongly nonlinear interfacial model the same resistance on a model vessel of FRAM’s dimensions. The
intersection between the empirical and computed resistance curves provides a rather
accurate estimate of the ship speed. This is then studied as function of the depth of the
pycnocline, a quantity that was not measured when the phenomenon was observed.
The wave wake at the small range of ship speeds is evaluated.
Ship internal wave wakes have commonly been analysed by linear theory assuming a source, pressure or thin-ship representation [3–6]. The dead water resistance
due to the internal wave wake on a semi-submerged slender prolate spheroid was
calculated in the linear regime comparing to experiments [7, 8]. Nonlinear models of the dead water problem have been requested because of the poor comparison
between field measurements and linear theory [9]. The same lack was expressed in
relation to two-dimensional model tank experiments [10].
Strongly nonlinear analysis and calculations have recently been developed for the
dead water problem by Grue [11] and Grue et al. [12]. The main point is that the
internal wave wake becomes rather prominent when the ship draught, here denoted
by b 0 , is comparable to the mid-depth of the pycnocline, here denoted by h 0 , i.e.
b 0 ∕h 0 ∼ 1. In the subcritical range, the nonlinear dead water resistance deviates from
the linear theory by a large amount, when the Froude number Fr defined in (2) is less
than 0.85. The differences are essential also for 0.85 < Fr < 0.95. They are found to
be minor at the critical speed. However, at Fr = 1 the nonlinear and linear wave fields
differ fundamentally [11]. At large supercritical speeds there are essential differences
between nonlinear and linear representation of the wave wake and the dead water
resistance, where the nonlinear calculations for a large cargo ship cruising along
a stratified subarctic fjord compared very well to a set of field observations, while
the linear predictions were inferior [12]. Recent three-dimensional experiments with
a bluff body in a moderately wide tank support the nonlinear calculations of the
colossal internal wave resistance at the small Froude numbers [13].
The present nonlinear calculations very well represent the wave wake and internal
wave resistance at the very small subcritical Froude numbers in question. The linear
calculations exhibit rather the contrary where an observed asymmetry of the interfacial elevation at the ship is not found. Further, the linear wave wake has a vanishingly
small amplitude. The dead water resistance is almost zero.
The paper is organised as follows: section “Nansen’s Observations of the Dead
Water” describes Nansen’s observations of the dead water. The dead water resistance is calculated from the observations. The nonlinear interfacial method is outlined in section “Nonlinear Interfacial Model”, and the internal wave resistance force
obtained in section “Dead Water Resistance”. The intersection between the empirical and computed resistance as function of the Froude number is then discussed. This
J. Grue
Ekman [2]. Our purpose is to calculate as exactly as possible the internal wave (dead
water) resistance, the wave wake and speed of the FRAM, for the conditions in the
field. The computations illustrate and enhance the accuracy of the observations.
We first derive from the empirical descriptions of Nansen [1] and Ekman [2] the
dead water resistance on FRAM. Then we compute by a strongly nonlinear interfacial model the same resistance on a model vessel of FRAM’s dimensions. The
intersection between the empirical and computed resistance curves provides a rather
accurate estimate of the ship speed. This is then studied as function of the depth of the
pycnocline, a quantity that was not measured when the phenomenon was observed.
The wave wake at the small range of ship speeds is evaluated.
Ship internal wave wakes have commonly been analysed by linear theory assuming a source, pressure or thin-ship representation [3–6]. The dead water resistance
due to the internal wave wake on a semi-submerged slender prolate spheroid was
calculated in the linear regime comparing to experiments [7, 8]. Nonlinear models of the dead water problem have been requested because of the poor comparison
between field measurements and linear theory [9]. The same lack was expressed in
relation to two-dimensional model tank experiments [10].
Strongly nonlinear analysis and calculations have recently been developed for the
dead water problem by Grue [11] and Grue et al. [12]. The main point is that the
internal wave wake becomes rather prominent when the ship draught, here denoted
by b 0 , is comparable to the mid-depth of the pycnocline, here denoted by h 0 , i.e.
b 0 ∕h 0 ∼ 1. In the subcritical range, the nonlinear dead water resistance deviates from
the linear theory by a large amount, when the Froude number Fr defined in (2) is less
than 0.85. The differences are essential also for 0.85 < Fr < 0.95. They are found to
be minor at the critical speed. However, at Fr = 1 the nonlinear and linear wave fields
differ fundamentally [11]. At large supercritical speeds there are essential differences
between nonlinear and linear representation of the wave wake and the dead water
resistance, where the nonlinear calculations for a large cargo ship cruising along
a stratified subarctic fjord compared very well to a set of field observations, while
the linear predictions were inferior [12]. Recent three-dimensional experiments with
a bluff body in a moderately wide tank support the nonlinear calculations of the
colossal internal wave resistance at the small Froude numbers [13].
The present nonlinear calculations very well represent the wave wake and internal
wave resistance at the very small subcritical Froude numbers in question. The linear
calculations exhibit rather the contrary where an observed asymmetry of the interfacial elevation at the ship is not found. Further, the linear wave wake has a vanishingly
small amplitude. The dead water resistance is almost zero.
The paper is organised as follows: section “Nansen’s Observations of the Dead
Water” describes Nansen’s observations of the dead water. The dead water resistance is calculated from the observations. The nonlinear interfacial method is outlined in section “Nonlinear Interfacial Model”, and the internal wave resistance force
obtained in section “Dead Water Resistance”. The intersection between the empirical and computed resistance as function of the Froude number is then discussed. This