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The Special Form of Second Order Accuracy N-S Equations for Incompressible Fluid

Classic N-S equation has first order accuracy in both of time and space. It has only the terms of first order, without the terms of second or higher order. These terms are relative in time and space steps. The time and space steps, as basic elements of fluid research, should be only some finite quantities and not be infinitely near to zero as defined in mathematics. If the terms of second or higher order can be ignored depends on the value of the corresponding derivative multiplied. Compared with terms of first order, the terms of second or higher order can be ignored under the condition of laminar flow. However, under the condition of turbulent flow, these can’t be ignored yet. When turbulent flow develops fully, the terms of first order, compared with terms of second order, can be ignored. So, it is why classic N-S equations aren’t closed when they are used to analyze turbulent flow. On the basic, many different special forms of the second order accuracy N-S equations of incompressible fluid are derived.