## Laminar wake ﬂow behind a hump on a wall

##### Date

2018

##### Authors

Julyan, Jonathan

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##### Abstract

The laminar wake ﬂow behind a hump on a solid wall boundary is investigated. A Blasius boundary layer ﬂow is perturbed by the hump and a wake forms directly downstream. Triple deck theory is applied to the wake region and the ﬂow is divided into three decks. The governing equations are derived for each deck for both the near and the far wake. Particular attention is paid to the role of the boundary layer displacement eﬀect. The conservation laws and conserved quantities for the governing equations are derived. The multiplier method is applied to the linearised governing equations for small humps and a basis of conserved vectors is constructed. Since, in general, the problem contains an unknown non-homogeneous boundary condition, each conserved vector needs to be carefully chosen and additional restrictions need to be applied to ensure that each conserved quantity, which is obtained by integrating the corresponding conservation law across the wake and imposing the relevant boundary conditions, has a ﬁnite value. Four non-trivial conserved quantities are found; three of which have only now been identiﬁed. The four conserved quantities relate to the conservation of mass, drag and the ﬁrst and second moments of the momentum deﬁcit. For each case the existence of a solution that satisﬁes the governing equations, boundary conditions and a ﬁnite valued conserved quantity is discussed. The solution corresponding to the near wall-wake ﬂow is further discussed. Although the far wall-wake does not satisfy a conserved quantity, for completeness, it is included in this work.

##### Description

The laminar wake ﬂow behind a hump on a solid wall boundary is investigated. A Blasius boundary layer ﬂow is perturbed by the hump and a wake forms directly downstream. Triple deck theory is applied to the wake region and the ﬂow is divided into three decks. The governing equations are derived for each deck for both the near and the far wake. Particular attention is paid to the role of the boundary layer displacement eﬀect. The conservation laws and conserved quantities for the governing equations are derived. The multiplier method is applied to the linearised governing equations for small humps and a basis of conserved vectors is constructed. Since, in general, the problem contains an unknown non-homogeneous boundary condition, each conserved vector needs to be carefully chosen and additional restrictions need to be applied to ensure that each conserved quantity, which is obtained by integrating the corresponding conservation law across the wake and imposing the relevant boundary conditions, has a ﬁnite value. Four non-trivial conserved quantities are found; three of which have only now been identiﬁed. The four conserved quantities relate to the conservation of mass, drag and the ﬁrst and second moments of the momentum deﬁcit. For each case the existence of a solution that satisﬁes the governing equations, boundary conditions and a ﬁnite valued conserved quantity is discussed. The solution corresponding to the near wall-wake ﬂow is further discussed. Although the far wall-wake does not satisfy a conserved quantity, for completeness, it is included in this work.