Pagrindinis šio darbo tikslas - sudaryti efektyvius skaitinius algoritmus elipsinio tipo uždaviniams spręsti. Galiorkino metodu sprendžiami du uždaviniai: vienmatis pasienio sluoksnio uždavinys ir dvimatis elipsinis uždavinys su L formos geometrijos sritimi. Algoritmų efektyvumui gerinti naudojami adaptyvieji tinklai, sudaryti remiantis aposterioriniais įverčiais. Darbe parodyta kaip iš dualiųjų įverčių teorijos gauti Bakhvalovo tinklą. Taip pat parodytas Šiškino tinklo ryšys su Bakhvalovo tinklu. Iš aposteriorinių įverčių teorijos gautos σ parametro (jis naudojamas Bakhvalovo ir Šiškino tinkluose) reikšmės, kurios skiriasi skirtinguose normose. Teorinės σ reikšmės patvirtintos skaitiniais eksperimentais, jas galima naudoti kaip rekomendacija sudarant Šiškino arba Bakhvalovo tinklus. Remiantis aposterioriniais įverčiais sudaryta Šiškino tinklo modifikacija, kuri prisitaiko prie šaltinio ir reakcijos funkcijų ypatumų, atlikti skaitiniai eksperimentai. Aposteriorinių įverčių metodas pritaikytas dvimačiam uždaviniui, atlikti eksperimentai, ištirtas adaptyviojo tinklo efektyvumas. / The objective of this paper is to construct effective numerical algorithms for elliptic problems. We use Galerkin method to solve two problems: a one-dimensional boundary layer problem and the two-dimensional elliptic problem with a specific geometry of L form. To optimize computations we use adaptive meshes that are constructed from aposteriori error estimates. We show how to derive Bakhvalov mesh from aposteriori estimates. Also we show the relation between Shishkin and Bakhvalov meshes. From aposteriori estimates we derive the exact values of σ parameter(which is used in Shishkin and Bakhvalov meshes), which depends on a norm in which we calculate the error. Theoretical σ values were confirmed by calculations, they can be used as a recommendation, when a problem is being solved using Shishkin or Bakhvalov meshes. Also we use duality-based aposteriori error estimation to construct a modification of Shishkin mesh, which use additional information about parameters in differential equation, we experimentally compare this mesh with the original(Shishkin) one. We apply aposteriori error estimation technique to a two-dimensional problem and investigate efficiency of an adaptive mesh.

Despite years of research in high speed boundary layer flow, there is still a need for insightful experiments to realize key features of the flow like boundary layer response to different conditions and related transition mechanisms. Volumes of data on the these problems point to the fact that there is still much to be understood about the nature of boundary layer instability causing transition and growth of boundary layer in different conditions. Boundary layer stability experiments have been found to be more useful, in which the boundary layer is perturbed and its behavior observed to infer useful conclusions. Also, apart from the stability part, the effect of various changes in boundary layer due to the perturbation makes interesting observation to gain more insight into the understood and the not so understood facets of the same. In view of the above, the effect of a steady axisymmetric thermal bump is investigated on a hypersonic boundary layer over a 60º sharp cone cylinder model. The thermal bump, placed near tip of the cone, perturbs the boundary layer, the behavior of which is observed by recording the wall heat flux on the cone and cylinder surface using platinum thin film sensors. The state of the boundary layer is qualitatively assessed by the wall heat flux comparisons between laminar and turbulent values. The same thermal bump also acts as a heat addition source to boundary layer in which case this recorded data provides a look into the effect of the heat addition to the wall heat flux. To gain a larger view of heat addition causing changes to the flow, effects of change in enthalpy are also considered. Experiments are performed in the IISc HST2 shock tunnel facility at 2MJkg−1 stag-nation enthalpy and Mach number of 8,with and without the thermal bump to form comparisons. Some experiments are also performed in the IISc HST3 free piston driven shock tunnel facility at 6MJkg−1, to investigate the effect of change in stagnation enthalpy on the wall heat flux. To support the experimental results theoretical comparisons and computational studies have also been carried out. The results of experiments show that the laminar boundary layer over the whole model remains laminar even when perturbed by the thermal bump. The wall heat flux measurements show change on the cone part where there seems to be fluctuation in the temperature gradients caused by the thermal bump, which decrease at first and then show an increase towards the base of the cone. The cylinder part remains the same with and without the thermal bump, indicating heavy damping effects by the expansion fan at cone cylinder junction. A local peak in wall heat flux is observed at the junction which is reduced by 64% by the action of the thermal bump. The possible reason for this is attributed to the increased temperature gradients at the wall due to delayed dissipation of heat that is accumulated in the boundary layer as a result of the thermal bump action. The comparison of data for enthalpies of 2MJkg−1 and 6MJkg−1 show that there are negligible real gas effects in the higher enthalpy case and they do not affect the wall heat flux much. Also it is found that the thermal bump fails to dump heat into the flow directly though it creates heat addition virtually by mere discontinuity in the surface temperature and causes temperature gradients fluctuation in the boundary layer. Considering the thermal bump action and the change in stagnation enthalpy of the flow, there seems to be no change in both cases that can be attributed to a common observation resulting from the factor of change in heat inside the boundary layer.