Hyponormal operator

Flow forces according to the Morison equation for a body placed in a harmonic flow, as a function of time. Blue line: drag force; red line: inertia force; black line: total force according to the Morison equation. Note that the inertia force is in front of the phase of the drag force: the flow velocity is a sine wave, while the local acceleration is a cosine wave as a function of time.

In fluid dynamics the Morison equation is a semi-empirical equation for the inline force on a body in oscillatory flow. It is sometimes called the MOJS equation after all four authors—Morison, O'Brien, Johnson and Schaaf—of the 1950 paper in which the equation was introduced. The Morison equation is used to estimate the wave loads in the design of oil platforms and other offshore structures.

Description

File:Jacket Wave.ogv The Morison equation is the sum of two force components: an inertia force in phase with the local flow acceleration and a drag force proportional to the (signed) square of the instantaneous flow velocity. The inertia force is of the functional form as found in potential flow theory, while the drag force has the form as found for a body placed in a steady flow. In the heuristic approach of Morison, O'Brien, Johnson and Schaaf these two force components, inertia and drag, are simply added to describe the force in an oscillatory flow.

The Morison equation contains two empirical hydrodynamic coefficients—an inertia coefficient and a drag coefficient—which are determined from experimental data. As shown by dimensional analysis and in experiments by Sarpkaya, these coefficients depend in general on the Keulegan–Carpenter number, Reynolds number and surface roughness.

The descriptions given below of the Morison equation are for uni-directional onflow conditions as well as body motion.

Fixed body in an oscillatory flow

In an oscillatory flow with flow velocity $u(t)$, the Morison equation gives the inline force parallel to the flow direction:

$F=\underset{F_I}{\underbrace{\rho C_{m}V\dot{u}}}+\underset{F_D}{\underbrace{\frac{1}{2}\rho C_{d}Au\left|u\right|}},$

where

• $F(t)$ is the total inline force on the object,
• $\dot{u}\equiv \text{d}u/\text{d}t$ is the flow acceleration, i.e. the time derivative of the flow velocity $u(t),$
• the inertia force $F_I=\rho C_{m}V\dot{u}$, is the sum of the Froude–Krylov force $\rho V\dot{u}$ and the hydrodynamic mass force $\rho C_{a}V\dot{u},$
• the drag force $F_D=\frac{1}{2}\rho C_{d}Au\left|u\right|$,
• $C_m=1+C_a$ is the inertia coefficient, and $C_a$ the added mass coefficient,
• A is a reference area, e.g. the cross-sectional area of the body perpendicular to the flow direction,
• V is volume of the body.

For instance for a circular cylinder of diameter D in oscillatory flow, the reference area per unit cylinder length is $A=D$ and the cylinder volume per unit cylinder length is $V=\frac{1}{4}\pi D^2$. As a result, $F(t)$ is the total force per unit cylinder length:

$F=C_m\rho \frac{\pi }{4}{D}^2\dot{u}+C_d\frac{1}{2}\rho Du\left|u\right|.$

Besides the inline force, there are also oscillatory lift forces perpendicular to the flow direction, due to vortex shedding. These are not covered by the Morison equation, which is only for the inline forces.

Moving body in an oscillatory flow

In case the body moves as well, with velocity $v(t)$, the Morison equation becomes:

$F=\underset{a}{\underbrace{\rho V\dot{u}}}+\underset{b}{\underbrace{\rho C_{a}V(\dot{u}-\dot{v})}}+\underset{c}{\underbrace{\frac{1}{2}\rho C_{d}A(u-v)|u-v|}}.$

where the total force contributions are:

• a: Froude–Krylov force,
• b: hydrodynamic mass force,
• c: drag force.

Note that the added mass coefficient $C_a$ is related to the inertia coefficient $C_m$ as $C_m=1+C_a$.

Limitations

• The Morison equation is a heuristic formulation of the force fluctuations in an oscillatory flow. The first assumption is that the flow acceleration is more-or-less uniform at the location of the body. For instance, for a vertical cylinder in surface gravity waves this requires that the diameter of the cylinder is much smaller than the wavelength. If the diameter of the body is not small compared to the wavelength, diffraction effects have to be taken into account.
• Second, it is assumed that the asymptotic forms: the inertia and drag force contributions, valid for very small and very large Keulegan–Carpenter numbers respectively, can just be added to describe the force fluctuations at intermediate Keulegan–Carpenter numbers. However, from experiments it is found that in this intermediate regime—where both drag and inertia are giving significant contributions—the Morison equation is not capable to describe the force history very well. Although the inertia and drag coefficients can be tuned to give the correct extreme values of the force.
• Third, when extended to orbital flow which is a case of non uni-directional flow, for instance encountered by a horizontal cylinder under waves, the Morison equation does not give a good representation of the forces as a function of time.

See also

• Drag equation

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, 530 pages